HOMEWORK ANSWERS (ODDS)
Math A Honors
Module #6
2015 - 2016
Created in collaboration with
Utah Middle School Math Project
A University of Utah Partnership Project
San Dieguito Union High School District
6.0A Review: Geometry Vocabulary
Name:
Period:
Boxes that are blank under ―Symbolic Name‖ are because there is not an established symbol for the term.
Term
Definition
Symbolic Name
Example
Point
A location in space.
It has no size.
Point A
A
Line
A series of points
that extend in
opposite directions
without end.
Line Segment
Ray
Plane
Collinear
Non-collinear
Part of a line. It
has two endpoints.
Part of a line with
exactly one
endpoint. It is
named endpoint
first.
A flat surface with
no thickness. It
extends without
end in all
directions. Named
by 3 points.
n
⃡
or ⃡
B
or n
A
̅̅̅̅ or ̅̅̅̅
represents the
length
Q
P
D
or
E
X
Y
E
Plane ABC
or Plane E
Points that lie on
the same line.
Points that cannot
be contained by
one line.
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
B
A
C
C
A
B
B
C
A
2
B
Parallel Lines
Intersecting
Lines
Lines that lie in the
same plane that do
not intersect.
⃡
⃡
D
A
Lines that lie in the
same plane that
cross.
C
D
C
A
E
B
Skew Lines
E
Lines that do not lie
in the same plane.
They are not
parallel and do not
intersect.
F
D
H
G
C
B
A
B
Midpoint
Angle
A point that is the
middle of a line
segment.
Two rays that meet
at a common
endpoint form an
angle.
AX = XB
or
̅̅̅̅ ̅̅̅̅
or
X
A
or
A
B
Vertex
The common
endpoint of an
angle.
Sides
The two rays that
form an angle are
also known as the
sides of the angle.
C
Point B
See picture for angle.
or
See picture for angle.
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D
Congruent
Angles
T
Angles that have
the same measure.
O
C
G
A
Y
Congruent
Segments
Segments that
have the same
measure.
̅̅̅̅ ̅̅̅̅
AB = XY
X
A
B
Protractor
The tool used to
measure an angle
in degrees.
C
Perpendicular
Lines
Two lines that
intersect and form
a right angle.
⃡
E
⃡
A
Acute Angle
An angle that
measures between
0 and 90 .
Obtuse Angle
An angle that
measures between
90 and 180 .
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
B
D
4
Right Angle
An angle that
measures
.
Straight Angle
An angle that
measures 180 .
Complementary
Angles
A pair of angles
whose sum is 90 .
They do not need
to be adjacent.
S
U
V
T
Y
Supplementary
Angles
A pair of angles
whose sum is 180 .
X
W
Adjacent Angles
Angles that share a
vertex and a side
but no points in
their interiors.
A
C
Z
D
B
Polygon
A plane figure
formed by line
segments, having
at least three sides.
The sides meet
only at their
endpoints.
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Regular Polygon
A polygon with all
sides and angles
congruent.
U
Triangle
A polygon with
three sides.
or
S
Acute Triangle
A triangle with all
acute angles.
Obtuse Triangle
A triangle with one
obtuse angle.
Right Triangle
A triangle with one
right angle.
Scalene Triangle
A triangle with no
sides congruent.
Isosceles
Triangle
N
A triangle with at
least two congruent
sides.
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Equilateral
Triangle
A triangle with
three congruent
sides.
B
Quadrilateral
A polygon with four
sides.
☐ABCD
C
D
A
Congruent
The same shape
and the same size.
Congruency
Marks
Tick marks on sides
or angles that have
the same measure
See marks on some of the diagrams in this
table.
Extra boxes in case teachers would like to
include other definitions.
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6.0B Review: Angle Classification and Using a Protractor*
Name:
Period:
You will likely need to review how to use a protractor with your students for this activity. Here’s a Learn Zillion
video that does this:
https://learnzillion.com/lessons/2973-measure-angles-to-the-nearest-degree-with-protractors
Review Angles: For each angle below, a) use a protractor to find the measurement in degrees and b) write its
classification (acute, obtuse, right or straight.)
1.
2.
Degrees:
Degrees:
Classification: Right angle
Classification: Acute angle
3.
4.
Degrees: 100
Degrees: 70
Classification: Obtuse angle
Classification: Acute angle
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5.
6.
Degrees:
Degrees:
Classification: Straight angle
Classification: Right angle
7.
8.
Degrees:
Classification: Obtuse angle
Degrees:
Classification: Acute angle
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6.1A Homework: Triangle Practice*
Name:
Period:
Find the sum of the angles for the following polygons beginning with the proper formula.
1. Pentagon
2. Octagon
3. 15-sided polygon
(pentadecagon)
540
2340
is a scalene acute triangle. If
and
, what is
? Show your work.
For #5-8, determine whether it is possible to form a triangle with the three given side lengths. Explain
why or why not using the Triangle Inequality Theorem.
4.
1, 4, 10
5.
No, a triangle is not possible because
1.5 + 3.5 5.
6.
4, 3, 6.9
7.
Yes, it is possible to form a triangle with these
three side lengths because
.
8. Mrs. Turnip wants to build a vegetable garden. She has three pieces of fencing. One is 8 feet long, one is
5 feet, and the other is 3 feet long. Can she build a closed triangular garden with these three boards,
provided she is not allowed to re-cut any of the board pieces or have any pieces left over? Include a
diagram with your explanation.
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9. Anna used the scale on a map to calculate the
distances “as the crow flies” (meaning the
perfectly straight distance) from three points in
Central America and the Caribbean islands,
marked on the map to the right.
a. According to Anna, how far is it from Jamaica
to Panama if you go directly?
700 miles
b. According to Anna, how far is it from Jamaica
to Panama if you do go through Honduras
first?
600 miles
c. Another way of stating the Triangle Inequality
Theorem is “the shortest distance between
two points is a straight line.” Explain why Anna
must have made a mistake in her calculations.
10. Two side lengths of a triangle are 5 cm and 7 cm.
a. What is the smallest possible whole number length of the third side? Justify your answer.
b. What is the smallest possible length of the third side to the nearest
of a centimeter?
11. Two side lengths of a triangle are 6 cm and 8 cm.
a. What is the largest possible whole number length of the third side? Justify your answer.
13 cm—note that at 14 the other two sides would fall flat so the largest INTEGER value is 13. Again,
take time to explore 11.5, 11.9, 11.99, etc.
b. What is the largest possible length of the third side to the nearest of a centimeter?
13 cm—note that at 14 the other two sides would fall flat so the largest INTEGER value is 13. Again,
take time to explore 11.5, 11.9, 11.99, etc.
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For the next two problems:
(a) Carefully duplicate the triangle using a ruler and protractor. Label the sides lengths (rounding to the
nearest tenth of a cm) and angle measures (rounding to the nearest whole number of a degree) for your new
triangle. (b) Then write an inequality that shows that the Triangle Inequality Theorem holds. (c) Classify the
triangle based on its sides and angles.
12.
a. Copy and label the triangle:
A
B
C
b. Triangle Inequality Theorem statement:
c. Classify the triangle based on its sides and
angles:
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13.
a. Copy and label the triangle as in #13:
E
D
F
b. Triangle Inequality Theorem statement:
example
.
c. Classify the triangle based on its sides and
angles:
Scalene acute triangle
True or false. Explain your reasoning with a specific example to illustrate your explanation.
14. An acute triangle has three sides that are all different lengths.
15. A scalene triangle can be an acute triangle as well.
True, the sum of 3 different acute angles(which means their opposite sides are different lengths) can equal
180°, forming a triangle.
16. A scalene triangle has three angles less than 90 degrees.
17. A triangle with a 100° angle must be an obtuse triangle.
True, a triangle with 1 obtuse angle is an obtuse triangle.
18. The angles of an equilateral triangle are also equal in measure.
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Spiral Review:
20. Find 45% of 320 without a calculator. Show your
work.
19. Simplify
21. Evaluate
18
for
22. Kim, Laurel, and Maddy are playing golf. Kim
ends with a score of 8. Laurel’s score is 4.
Maddy scores +5. What is the difference between
the scores of Maddy and Kim?
Define a variable, write an equation, solve it, and answer in a complete sentence.
23. The temperature increased 2º per hour. If the temperate rose 15 , how many hours had the temperature
been rising?
Let x = the number of hours
The temperature rose for 7.5 hours.
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6.1B Homework: Building Triangles Given Three Measurements*
Name:
Period:
In #1-2, list the angles of the triangle from
smallest to largest without using a protractor.
Triangles are not drawn to scale.
2.
1.
B
B
7
8
A
10°
8
50°
9
A
10
11
C
In #3-4, list the sides of the triangle from shortest to
longest without using a ruler. Triangles are not
drawn to scale.
̅̅̅̅ , ̅̅̅̅, ̅̅̅̅
4.
3.
C
B
B
A
A
40°
30°
C
C
5. Use the grid below to draw and label
with AB = 5 units, BC = 7 units, and
.
A
B
6. Is
the included angle to the given sides?
Explain.
C
7. What is the area of the triangle in #5? Justify your
answer.
17½ units: (7*5)/2
8. Will all your classmates who draw a triangle like #5 get a triangle with the same area? Explain.
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9. Which triangle(s) below have two sides 5 and 8 units, and a non-included angle of 20 adjacent to the side
of length 8? Both of them.
10. Are the two triangles in #9 exactly the same and size and shape? Explain why or why not.
11. Paul lives 2 miles from Rita. Rita lives 3 miles from the shopping mall. What are the shortest and longest
straight-line distances Paul could live from the mall? All three houses are not in a straight line. Sketch a
picture to justify your work.
Paul could be 2 mile or 4 miles away.
For #12, define your variable, write an algebraic equation, solve, and answer in a complete sentence.
12. Three segments have a total length of 26 inches. The first segment is two less than three times the
second segment. The third segment is one more than five times the second segment. What are the
lengths of the three segments? Can a triangle be formed with these segments? Why or why not?
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For #13-16, decide whether there are zero, one, or more than one possible triangles with the given
conditions. Use either a ruler and protractor OR graph paper strips and protractor to draw the
triangles.
13. A triangle with sides that measure 5, 12, and 13
cm.
How many possible triangles can be formed?
One
If possible, what kind of triangle is formed? If not
possible, state why not.
Right scalene triangle.
14. A triangle with angles 100° and 20°, and an
included side of 2 cm.
How many triangles can be formed?
If possible, what kind of triangle is formed? If not
possible, state why not.
Labeled picture:
Labeled picture:
15. A triangle with two sides of 5 and 7 cm, and an
included angle of 45°.
16. A triangle HIJ in which m ∠HIJ = 60°,
m ∠JHI = 90°, and m ∠IJH = 55°.
How many triangles can be formed?
One
How many possible triangles can be formed?
If possible, what kind of triangle is formed? If not
possible, state why not.
Right isosceles triangle
If possible, what kind of triangle is formed? If not
possible, state why not.
Labeled picture:
Labeled picture:
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Spiral Review:
Solve. Show all algebraic steps.
17.
m=6
18.
19. Write as a percent and decimal.
60%, 0.
20. Show how to simplify the following expression with a number line –6 + (–3)
21. Kurt earned $550 over the summer. If he put 70% of his earnings into his savings, how much money did
he have left to spend? Answer in a complete sentence.
Kurt has $165 left over.
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Section 6.1 Review*
Name:
1.
Period:
is an isosceles obtuse triangle. If
28 and 124
, what are the measures of the other angles?
Given the following criteria, construct a triangle using a protractor and a ruler:
2. Draw and label
with
3. Draw and label
with
4. Draw and label
with
AB = 5 units, BC = 7 units, and
,
,
, CA = 4 units,
CA = 10 units.
and BA = 6 units.
and
.
See student responses.
Determine if the following side lengths will make a triangle. Show your work using the Triangle
Inequality Theorem to justify your answer.
5. 2 in, 5in, 7in
6. 33 mm, 93.2 mm, 70 mm
7. 1 ft, 2.7 ft, 7 ft
No, 2 + 5 7
No, 1 + 2.7 7
8. 5.1 in, 3.24 in, 8.25 in
9.
cm,
cm, 8 cm
10. 6.01 cm, 5 cm, 11.22 cm
No,
11. Your friend is having a hard time understanding how angle measures of 30°, 60°, 90° might create more
than one triangle. Draw two different triangles that have those angle measures and explain why the two
triangles are different.
Encourage students to use GeoGebra or a protractor and ruler. It may help to have students create angles
of
and then create the triangles from there. It will also be very helpful if students create
their triangle on grid paper. You will use this activity in 6.2A Lesson. Discuss with students that even
though the two triangles have the same angle measures, the side’s lengths are not the same. In the next
section we will begin the discussion of scaling triangles (similar triangles).
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Determine if the given information will make a unique triangle. Explain why or why not.
13. Angles 73°, 7°, 100°
14. Angles 80° and 25° and
12. Side lengths 3 and 5 and an
included angle of 67°
No, this does not create a
included side of 12
unique triangle. Given 3
angles the triangles are similar,
but not necessarily congruent.
For #15-16, determine if the given angles can make a triangle. Explain why or why not.
15. Angles 25°, 70°, 95°
16. Angles 30°, 40°, 20°
No the angles add to 190 degrees, and they
should add to 180 degrees.
17. In
, AB = 12, BC = 7, AC = 9. List the
angles from least to greatest. (Sketch and label a
picture.)
18. In
,
,
,
.
List the sides from greatest to least. (Sketch and
label a picture.)
19. A triangle has two angles with measures 60 and 75 and an included side measuring 20 centimeters.
Which figure(s) represents this triangle?
a.
b.
c.
d.
20. If you drew a triangle with these measurements, which kind of triangle would it be?
Angles
40
50
90
Sides
4 cm
7 cm
8 cm
a. Acute scalene triangle
b. Right scalene triangle
c. Acute isosceles triangle
d. Right equilateral triangle
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Spiral Review
21. Find the product:
22. Without a calculator, what percent of 80 is 60?.
Simplify the following:
23.
24.
25. Order the numbers from least to greatest.
26. Solve.
-2.15,
17
, 2.7, - 2.105
7
-2.15, -2.105,
, 2.7
27. Given the following table, find the indicated unit rate:
___15__ push-ups per day
Days Total Push-ups
2
30
4
60
29
435
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6.2A Homework: Comparing the Perimeter and Area of Polygons*
Name:
Period:
Below is a table that describes the dimensions, perimeter, area and change of side lengths for a rectangle
that’s scaled in different ways. In the first row you find that the 5x20 rectangle has a perimeter of 50 and an
area of 100; there is no change in side length here because it’s the original rectangle. In the next row the
dimensions become 10x40 giving it a perimeter of 100 and area of 400; the side length change is twice the
original. Use this information to do the following:



Examine the completed rows to understand what’s happening to the rectangle. Fill in any empty parts.
Graph the relationship between the perimeter (x-axis) and area (y-axis) on the graph below.
Describe the patterns you notice from the graph.
a)
Dimensions
Perimeter
Area
5 by 20
10 by 40
5/2 by 10
15 by 60
5/3 by 20/3
20 by 80
50
100
25
150
16.67
200
100
400
25
900
11.11
1600
Change of side
lengths from
original rectangle
Same
Twice
Half
Three times
One third
Four times
Change in
Perimeter
Change in Area
Same
Twice
Half
Three times
One third
Four times
Same
Four times
Fourth
9 times
One ninth
16 times
b)
c) Describe the patterns you notice:
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Define the variable. Write an equation. Solve and state your answer in a complete sentence.
1. A square has a perimeter of 25 inches. Find the length of a side.
6.25 inches.
2. A field has an area of 480,000 m2 and is 160 m wide. How much fencing is needed to enclose the field?
3. A brick walk 2 meters wide surrounds a rectangular pool that is 3 meters by 5 meters. Find the area of the
walk. Hint: Draw a picture.
48 m2.
Determine if the following statements are true or false. Explain your answer.
4. All isosceles triangles are similar.
5. All equilateral triangles are similar.
True, all three angles are the same and the sides
will be proportional.
6. All isosceles right triangles are similar.
7. All rectangles are similar.
False, rectangles are not similar to squares, a
special type of rectangle. The sides are not
proportional.
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Spiral Review:
8. What do we call triangles with all sides the same length?
9.
a. Katherine is visiting patients in a hospital. She visits 18 patients in 6 hours. At that rate, how many
patients will she visit in 9 hours? Show your work.
27 patients
b. What is Katherine’s unit rate?
10. What is an obtuse angle?
11. The price for two bottles of ketchup are given below:
A 20 oz. bottle of DELIGHT ketchup is 98¢ at the grocery store. A 38 oz. bottle of SQUEEZE ketchup is
$1.99 at the same grocery store.
a. Find the unit rate for each product
DELIGHT is $.049 per ounce and SQUEEZE is $.052 per ounce
b. What conclusions can you draw from this information?
DELIGHT is the least expensive of the two.
12. Simplify:
13. Simplify:
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6.2B Homework: Scaling Triangles*
Name:
Period:
1. Which scale would produce the smallest scale drawing of an object when compared to the actual object?
Explain your reasoning. C because 1 mm : 1 m is the same as 1 mm : 1000 mm, which produces the
smallest scale drawing.
a. 1 in : 10 in
b. 1 cm : 10 cm
c. 1 mm : 1m
2. The scale of a drawing is 6 cm : 1 mm. Is the scale drawing larger or smaller than the actual object?
Explain your reasoning.
3. A 0.035-centimeter-long paramecium appears to be 17.5 millimeters long under a microscope. What is
the power (scale factor or magnification scale) of the microscope? Show your work.
50
4. A model of an MC130E airplane has a scale factor of
.
a. If the width of the actual tail is 52 feet 8 inches, what is the width of the tail in the model?
b. If the height of the actual tail is 38 feet 5 inches, what is the height of the tail in the model?
5. The model of a dollhouse has been constructed using a scale of 1 to 48.
a. If the model’s door is two and one quarter inches high, how high is the actual dollhouse’s door?
108 inches tall, or 9 feet tall
b. If the model is 18 centimeters high, how tall is the actual dollhouse?
864 cm tall, or 8.64 m tall
c. What is the scale factor of the model of a doll house to the actual doll house?
48
d. How would the scale factor in part c compare to the scale factor of the actual doll house to the model?
Explain.
It would be , which is the reciprocal of the scale factor in part c.
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6. If the ratio of
answer.
to
is 3:5 and the length of ̅̅̅̅ is 6”, what is the length of ̅̅̅̅ ? Justify your
For #7-8, the pair of triangles given in each problem are proportional. The figures are not necessarily
drawn to scale.
Do the following:
 Solve for the unknowns by using proportions.
 State the scale factor between the two triangles.
 Express all answers exactly.
 Show your work.
7.
𝑥
𝑥
𝑢𝑛𝑖𝑡𝑠
𝑦
𝑦
𝑢𝑛𝑖𝑡𝑠
scale factor: 3
8.
5
x
8
1
6
9
y
Spiral Review:
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9. Suppose you were to flip a coin 3 times. What is the probability of getting heads all three times?
10. Samantha made 120 bracelets. She sells
How many bracelets does she have left?
of the bracelets and then decides to donate 50% of the rest.
11. Place each of the following integers on the number line below. Label each point:
B = –4
A=4
F
C
C = –15
D=7
B
12. Write 0.672 as a fraction and as a percent.
E = 18
F = –19
A
D
E
13. Simplify:
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6.2C Homework: Solve Scale Drawing Problems, Create a Scale Drawing*
Name:
Period:
Students will need a cm ruler.
The San Diego Trolley map below is not drawn to scale, but how off scale is it? Draw a triangle on the San
Diego Trolley map using the following locations as the vertices: Middletown, Fenton Parkway, and Euclid
Avenue.
Teacher note:
Student measure may
be slighty different
which will alter the
scales.
1. Measure the segment lengths and record them in the table below. Round your measurements to the
nearest tenth of a centimeter.
Segment
Distance (km)
Subway Map
Segment
Lengths (cm)
Map Scale
Middletown to Fenton Parkway
22.1 km
3.1 cm
1 : 7.1
Fenton Parkway to Euclid Avenue
27.9 km
4.5 cm
1 : 6.2
Euclid Avenue to Middletown
28.6 km
5.2 cm
1 : 5.5
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2. Is this portion of the map drawn to scale? Explain why or why not.
3. On a map, Breanne measured the straight-line distance between Los Angeles and San Francisco at 2
inches. The scale on the map is
1
inch = 43miles . What is the actual straight-line distance between Los
4
Angeles and San Francisco? Show all your work.
344 miles
4. What scale was used to enlarge the drawing below? How do you know?.
What is the scale factor? 2
5. On a separate sheet of grid paper, create the creature below so that it is a 1:3 enlargement of the original
model. Write your strategy for calculating lengths to the right of the picture.
This will likely take about 10 minutes.
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Spiral Review:
6. Suppose you flip a coin 3 times; what is the probability that you get heads exactly two times?
7. For each group of three side lengths, determine whether a triangle is possible. Write yes or no. Show
your work to justify your answer using the Triangle Inequality Theorem.
b.
c.
a.
Yes
Yes
8. Which number is greater: –24.41 or –24.4? Explain.
Solve.
10. Simplify:
9.
x=7
Define a variable. Write an algebraic equation. Solve and answer in a complete sentence.
11. A sweater costs $60.75 after the store increased the price 35%. How much did the sweater cost originally?
$45
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Section 6.2 Review*
Name:
Period:
1. Draw a scaled version of the following polygons given the scale factor.
a. Scale Factor: 2
b. Scale Factor:
A
A
13
12
24
26
B
B
10
5
C
C
The perimeter of an equilateral triangle is 45 cm. What is the length of each side?
2. A rectangle has length 18 and width 12. Find the perimeter.
3. Find the missing measurement in each problem
a. The scale factor from
to
BC is 3, what is the length of EF?
EF = 9 units
is 3. If
b. The scale factor from
to
OP is 30, what is the length of EL?
EL = 50 units
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
is . If
31
4. In each of the following, what is the scale factor that takes ABC to XYZ ?
a.
b.
5. Answer each of the following questions using proportional reasoning.
a. Becky is drawing a scale model of her
b. Carlos is reading a map of his town. The scale
neighborhood. She uses a scale of 1 in = 200
says 1 in = 4 miles. The distance from his
feet. If her block is 1000 feet long, what will be
house to the school is in. on the map. If
the length of the block on her scale model?
Carlos wants to walk to school from his house,
The length of the block will be 5 inches on her
how far will he have to walk?
model.
He will have to walk 1 ½ miles.
6. The dimensions of Romina’s rectangular garden are 2 feet by 3 feet. The dimensions of Santiago’s garden
are all quadrupled.
a. Find the perimeter and area of each garden.
b. Find the scale factor between the perimeters
of each garden.
c. Find the scale factor between the areas of
each garden.
d. In general, how are the scale factors of
perimeters and areas of scaled objects
related?
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
32
6.3A Homework: How Many Diameters Does it Take to Wrap Around a Circle?*
Name:
Period:
1. Complete the missing values in the table below. Round to the nearest tenth. Show all work in the given
space.
Radius
Diameter
Circumference
a.
16 in
32 in
100.5 in
b.
16.5 m
33 m
103.7 m
c.
25.0 cm
50.0 cm
157cm
2. What is the exact ratio of the circumference to the diameter of every circle?
3. If the radius of a circle is 18 miles:
a. What is the measure of the diameter?
36 miles
b. What is the measure of the circumference, exactly, in terms of pi?
36
c. What is the approximate measure of the circumference, to the nearest hundredth of a mile?
113.10 miles
4. For each of the three circles below, calculate the circumference. Express your answer both in terms of pi,
and also as an approximation to the nearest tenth.
Circumference
Diameter/Radius
Circumference
(in terms of )
Radius = 1.5 cm
Diameter = 5 cm
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
33
5. If the circumference of a circle is
feet, which of the following statements are true? Rewrite false
statements to make them true.
a. The circumference of the circle is exactly 62.8 feet.
False: circumference is approximately 62.8 ft
b. The diameter of the circle is 20 feet.
True
c. The radius of the circle is 20 feet.
False: the radius is 10 feet
d. The ratio of circumference : diameter of the circle is .
True
e. The radius of the circle is twice the diameter.
False: the radius is ½ the diameter or the diameter is twice the radius
6. The circumference of 2 objects is given. Calculate the diameter of each object, to the nearest tenth of a
unit.
Circumference
Diameter
Bottom of
Cupcake = 6.5 inches
Top of water
pail = 100 cm
7. The diameter or radius of 2 objects is given. Calculate the circumference of each object, to the nearest
hundredth of a unit.
Diameter/Radius
Circumference
Diameter of rim of the
drum = 24 inches
C = 75.40 in.
Radius of a table
top = 5.5 feet
C = 34.56 ft.
8. Three tennis balls are stacked and then tightly packed into a cylindrical can. Which is greater: the height
of the can, or the circumference of the top of the can? Justify your answer. (Hint: Draw a picture.)
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
34
9. A circular garden has a circumference of 43.96 yards. Leo is digging a straight-line trench along a
diameter of the garden at a rate of 7 yards per hour. How many hours will it take him to dig across the
garden?
a. Find the diameter. Round to the nearest whole number.
14 yards
b. Find the time it will take Leo to dig across the garden.
14/7 = 2 hours
It will take Leo 2 hours to dig across the garden.
10. The two coins shown at right have radii of 3 cm and 1 cm. The
smaller coin rolls along the circumference of the fixed larger coin
until the smaller coin returns to its original position. How many
revolutions has the small coin made?
11. A wire bent into the shape of a square encloses an area of 25 cm2. Then the same wire is cut in half and
bent into two identical circles. What is the radius of each circle? Round to the nearest hundredth.
1.59 cm
12. A circular path 2 feet wide has an inner diameter of 150 feet. How much farther is it around the outer edge
of the path than around the inner edge? Round to the nearest hundredth.
13. A gear on a bicycle has the shape of a circle. One gear has a diameter of 4 inches, and a smaller one has
the diameter of 2 inches. Justin says that the circumference of the larger gear is 2 inches more than the
circumference of the smaller gear. Do you agree? Explain your answer.
No, this is not correct because the circumference of the larger gear is 𝜋 and the circumference of the
smaller gear is 𝜋, and 𝜋
𝜋≈
inches. The circumference of the larger one is therefore 𝜋
inches longer, or about 6.28 inches longer.
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
35
14. Calculate the radius for each circle whose circumference is given in the table (the first entry is done for
you). Round to the nearest whole number. Then graph the values on a coordinate plane, with the radius
on the x axis and the approximate circumference on the y axis.
Radius of
circle in
inches
Circumference
of circle in
inches
4
≈
≈
≈
≈
≈
≈
≈
≈
How a Radius Relates to Circumference
15. Is the radius of a circle proportional to the circumference of the circle? Justify your answer.
Yes, the graph is a straight line that passes through the origin (a radius of zero would result in a
circumference of zero).
Spiral Review:
16. Factor the following expressions.
a.
–
b.
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
c.
–
36
17. Without using a calculator, determine which fraction is bigger in each pair. Justify your answer with a
picture or words.
a.
b.
18. Millie bought two sweaters for $30 each and three pair of pants for $25 each. She had a 20% off coupon
for her entire purchase. Model and write an expression for the amount of money Millie spent. State your
answer.
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
37
6.3C Homework: Area of a Circle*
Name:
Period:
1. Estimate the area of the circle at the right in
square units by counting.
≈
2. Use the formula for the area of a circle to
calculate the exact area of the circle above, in
terms of pi.
3. Round the answer to #2 to the nearest square unit. How accurate was your estimate in #1?
50 square units
4. Calculate the area of each circle below. Express your answer both exactly (in terms of pi) and
approximately, to the nearest tenth of a unit.
Area
Radius
Area
(in terms of )
5. A corner shelf is ¼ of a circle and has a radius of 10.5 inches. Find the area of the shelf. Round your
answer to the nearest hundredth. 86.59 square inches
10.5 in
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
38
6. The strongest winds in Hurricane Katrina extended 30 miles in all directions from the eye (center) of the
hurricane.
a. Draw a diagram of the situation.
b. What is the size of the area that felt the strongest winds?
1. Find the area of the shaded region at the right if the diameter of the
smaller circle is 6 cm and the shaded region is 10 cm wide. Round
to the nearest tenth.
The wall is 502.7 square centimeters
2. By calculating the areas of the square and the circle in the diagram below, determine how many times
larger in area the circle is than the square.
3. Draw a diagram to solve: A circle with radius 8 centimeters is enlarged so its radius is now 24 centimeters.
a. By what scale factor did the circumference increase? Show your work or justify your answer.
3 times
b. By what scale factor did the area increase?
9 times
c. Explain why this makes sense, using what you know about scale factor.
The scale factor of the area is always the square of the scale factor of the lengths.
4. How many circles of radius 1” could fit in a circle with radius 5” (if you could cut up and rearrange the area
of the circles of radius 1 in such a way that you completely fill in the circle of radius 5)? Show your work to
justify your answer.
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
39
5. The sides of a square field are 12 meters. A sprinkler in the center of the field sprays a circular area with a
diameter that corresponds to the side of the field. How much of the field is not reached by the sprinkler?
Round your answer to the nearest hundredth. (Hint: Draw a picture)
30.90 m2 of the field will not be reached by the sprinkler.
6. Two circles have the same radius. Is the combined area of the two circles the same as the area of a circle
with twice the radius? Explain.
7. The area of 2 objects is given. Calculate the radius of each object’s surface, to the nearest whole number.
Area
Radius
Area of a glass
porthole is 3.14 ft2
≈
ft
Area of a round area
rug is 153.86 ft2
≈
ft
8. A paraglider wants to land in the unshaded region in the square field
illustrated because the shaded regions (four quarter circles) are briar
patches. If the paraglider hits the field randomly due to unexpected
wind currents, what is the probability that she misses the briar patches?
Use the fact that the probability of landing in the unshaded region is the
area of the unshaded region divided by the total area of the field.
Round to the nearest whole percent.
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
40
Spiral Review:
Solve.
9.
10.
11. Simplify the expression in two different ways: 5(3 + 4)
Define a variable. Write an algebraic equation. Solve and answer in a complete sentence.
12. There are a total of 214 cars and trucks on a lot. If there are four more than twice the number of trucks
than cars, how many of both kinds of vehicles are on the lot?
13. Simplify:
–28
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
41
Section 6.3 Review*
Name:
Period:
1. Mr. Shay baked a giant cookie that is 16 inches in diameter. Mr. Anderson baked a cookie that is 1.5 feet
in diameter. Michelle and Katie are deciding which cookie is bigger.
a. Whose cookie is bigger? By how much?
Mr. Anderson’s cookie because it has an area of
in2, whereas Mr. Shay’s cookie has an area of
2
2
in . Mr. Anderson’s cookie is
in larger than Mr. Shay’s.
b. What is the ratio in area of the bigger cookie to the smaller cookie?
81:64
2. Solve each problem rounding to the nearest whole number. Show your work. Drawing a picture may be
helpful.
a. The radius of a circular pool is 10 feet. There
b. A covering for the pool costs $7 per square
is a border around the pool. The width of the
foot. About how much will the covering cost?
border is 5 feet. What is the area of the
border in square feet?
3. Solve each problem rounding to the nearest tenth. Show your work. Drawing a picture may be helpful.
a. The most popular pizza at Pavone’s Pizza is
b. What is the area of the largest circle that will fit
the 10-inch personal pizza with one topping.
in a square with an area of 64 square
What is the area of the pizza with a diameter
centimeters? Explain.
of 10 inches?
50.2 square centimeters
78.5 square inches
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
42
4. Use the given information to find the missing information. Give each answer in two forms: exact (in terms
of ) and rounded to the nearest hundredth unit.
a. Radius: 2 m
b. Diameter: 2 m
Circumference:
c. Radius: 5.5 in
Circumference:
d. Diameter: 40 in
Area:
e. Circumference: 69.08 cm
Diameter:
m, 6.28 m
Area:
f.
Area: 153.86 cm2
Radius:
Read each question carefully. Explain your reasoning. Round to the nearest whole number.
5. The radius of a large circle is 12 times the radius of the smaller circle.
a. If the radius of the smaller circle is 5 cm, what is the circumference of a smaller circle?
31.40 cm
b. What is the circumference of the larger circle?
376.99 cm
c. What is the ratio of the circumferences of the big circle to the small circle?
12:1
d. If the radius of the larger circle is 36 mm, what is the area of the larger circle?
4,071.50 square mm
e. What is the area of the smaller circle?
28.27 square mm
f.
What is the ratio of the area of the big circle to the small circle?
144:1
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
43
6.4A Homework: Special Angle Relationships*
Name:
1.
Period:
Find at least two examples of each angle relationship in the diagram. Name the angle pairs below, and
highlight the pairs of angles in the diagram, using a different color for each relationship.
a. Vertical angles
For example: ∠AHB & ∠FHG, ∠FHA & ∠GHB
b. Supplementary angles
For example: ∠AHB & ∠FHA, ∠AHF & ∠FHG
c. Complementary angles
For example: ∠EGD & ∠DGC, ∠HFG & ∠FHG
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
44
For each figure, calculate the three missing measures. Justify your answer.
2.
Angle
Measure of
angle
Justification
Angle
Measure of
angle
Justification
3.
150°
Vertical to ∠NOK
30°
Supplementary to ∠NOK
30°
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
Supplementary to ∠NOK or
vertical to ∠MOK
45
4.
For the figure below, fill in the missing measures. Justify your answers.
A
D
F
35°
E
52°
G
C
Justification
Measure
Angle
B
5.
Vertical to
Sum of the
measures of
∠FEC & ∠AEF
Refer to the figure below to complete the following statement:
63°
because it is complementary to ∠AZW
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
46
For #6-8, choose a correct answer. Sketch and label a diagram to justify your answer.
6.
If G is complementary to
7.
If
is
8.
If
is vertical
supplementary
, and
,
to
, and
to , and
then
must be:
, then
then
a. Obtuse
must be:
must be:
b. Acute
a. Obtuse
a. Obtuse
c. Right
b. Acute
b. Acute
c. Right
c. Right
For #9-11, write an algebraic equation, solve it, and complete the blanks below.
9.
10.
11.
C
G
C
4x - 15
A
H
F
6x - 25
B
6x
D
E
7x + 5
I
A
B
Equation:
Equation:
Equation:
x=
x=
x=
22
=
73
D
10x
7x - 5
=
E
5
=
40
For #12-14, before your variables, write and solve an algebraic equation. Answer in a sentence.
12.
Two angles are complementary, with one angle 24 greater than the measure of the other. Find the
measure of both angles.
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
47
13.
Two angles are supplementary. If the measure of one angle is 5 times the measure of the other, what
is the measure of each angle?
30 and 150
14.
If two vertical angles are supplementary, what is the measure of each angle?
Determine if the following statements are Always true, Sometimes true, or Never true. If sometimes or
never true, provide an example and a counterexample.
15.
Complementary angles have a sum of 180 . Never
16.
Vertical angles are complementary.
17.
Two supplementary angles with equal measure are right angles. Always
Find the measure of each lettered angle. Congruent angles and right angles are indicated.
18.
c
g
20°
d
h
65°
a
f
80°
b
e
95°
Spiral Review:
19.
Order the numbers from least to greatest. - ½ , ¼ , -¼, -1.2, -1.02, -.75
-1.2, -1.02, -.75, -½, -¼, ¼
20.
Find the quotient: -
21.
Find two unit rates for the statement below:
Izzy drove 357 miles on 10 gallons of gasoline.
22.
8 æ 3ö
¸ç- ÷
9 è 4ø
Convert the following units:
≈
feet = 37 inches
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
48
6.4B Homework: Exploring Circles
Name:
Period:
1. Suppose you have a circle with a radius of 1 unit. What is the area of the circle?
2. Suppose you have a circle with a radius of 2 units. What is the area of the circle?
3. What is the ratio of the area of circle 2 to circle 1?
4:1
4. What do you think the ratio of the areas would be for a circle of radius 3 units to circle 1? Explain.
5. What do you think the ratio of the areas would be for a circle of radius 4 units to circle 1?
16:1
6. What do you think the ratio of the areas would be for a circle of radius 1.5 units to circle 1?
7. If circle A has radius a and circle B has radius b, then the ratio of their circumference is . Explain.
Student answers will vary.
8. What is the ratio of their areas?
9. If the ratio of the radii of circle A and circle B is x, what is the ratio of their areas? Explain.
10. Complete the sentence: If you multiply the radius of a circle by y, then its area is multiplied by
.
11. What happens if you replace the word “circle” with “square” and the word “radius” with “side length” in
question numbers 3 – 7? Do your answers change?
The ratios remain the same. The answers will not change.
Extension:
12. A bathroom tissue company has manufactured a roll of tissue that is “twice as big” as a standard roll.
What will the roll look like compared to the original? Will it look twice as big?
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
49
Module 6 Review*
Name:
Period:
For #1-2, decide if the figures below are possible. Justify your conclusion with a mathematical
statement. To construct the triangles use:
 A ruler or strips of centimeter graph paper cut to the given lengths
 A protractor
 Optional: Teachers can let students use construction technology like GeoGebra
1. A triangle with angles that measure 20°, 70°, and
90°.
2. A triangle with sides 8 cm and 3 cm. The angle
opposite the 3 cm side measures 45°.
Possible or not? Why or why not?
Possible. The sum of the angle measures must
equal 180°, and 20 + 70 + 90 = 180
Possible or not? Why or why not?
If so, what kind of triangle? Draw and label.
Scalene right triangle
If so, what is the measure of the 3rd side? Draw
and label.
3. Two students were building a model of a car with an actual length of 12 feet.
a. Andy’s scale is
. What is the
b. Kate’s scale is
length of his model?
length of her model?
3 inches
6 inches
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
. What is the
50
4. At Camp Bright, the distance from the Bunk House to the Dining Hall is 112 meters. From the Dining Hall
to the Craft Building is 63 meters (in the opposite direction). The scale of the map for the camp is
0.5cm = 14meters . On the map:
a. What is the scaled distance between the Bunk
b. What is the scaled distance between the
House and the Dining Hall?
Dining Hall and the Craft Building?
If students are struggling, encourage them to draw a model. A length of 112 meters has a total of eight 14
meter units. Each 14 meter unit is ONE 0.5 cm on the scaled map. Thus the length on the scaled map is 4 cm.
5. In the similar L figures to the right,
a. What is the ratio of height of left figure : height of
right figure?
9:3 or 3:1
b. What is the reducing scale factor?
1/3
c. What is the ratio of area of left figure: area of right
figure?
36:4 or 9:1. The area scale factor will be the unit
scale factor squared.
6. Triangles ABC and RST are scale versions of
each other.
a. What is the scale factor from ABC to RST?
b. What is the scale factor from RST to ABC?
c. What is the distance between A and C?
d. What is the distance between R and S?
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
51
7. Redraw the figures at
right using the scale
factors below.
a. Use a scale factor
of 4 to re-draw the
square.
b. Use a scale factor
of ¼ to re-draw the
addition sign.
c. Use a scale factor
of 1.5 to re-draw
the division sign.
8. The Washington Monument is 555 feet and 5 1/8 inches tall. Bob wants to create a scale model that is 5
feet tall. What scale would you suggest Bob use for his model?
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
52
9. Calculate the circumference and area of the circles below. Express each dimension both exactly in terms
of pi, and as an approximation to the nearest hundredth of a unit.
C = 0.5 cm
C=6
C ≈ 1.57 cm
C ≈ 18.85 mm
A = 0.0625
sq. cm
A ≈ 0.19 sq. cm
A=9
mm
sq. mm
A ≈ 28.27 sq.mm
10. How many times would a circle with radius 4 units fit inside a circle with radius 12 units, if you could pack
the area tightly with no overlapping and no leftover space?
11. Are all circles similar? Justify your answer.
Yes, because the ratio between the diameter and circumference of every circle is π
12. Are all squares similar? Justify your answer.
13. Are all rectangles scaled versions of each other? Justify your answer.
No, ratios can differ. This is an important question. All regular figures and circles are scaled versions of
each other. Rectangles all have the same angles, but consecutive sides are not always in the same ratio.
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
53
14. A circle has an area of
your work.
. What is its circumference? Round to the nearest hundredth. Show
15. Find the missing angle measures for the figure below. Justify each answer.
∠MOL = 62°, vertical to ∠NOK
∠LON = 118°, supplementary to ∠NOK
∠KOM = 118°, supplementary to ∠NOK
16. Find all the missing angle measures for the figure below. Justify each answer.
D
A
F
103°
E
37°
C
G
B
17. Draw and label two intersecting lines for which ∠CDE and ∠ADR are vertical angles.
C
A
D
E
R
18. Draw and label two intersecting lines for which ∠HOG and ∠GOX are supplementary.
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
54
19. Draw and label ∠ABC and ∠DBE, a pair of complementary angles that are not vertical, adjacent, or
congruent.
D
E
70°
A
20°
B
C
20. Identify the shaded angle pairs diagrammed below as vertical, supplementary, or complementary.
21. Use the given relationship to find the missing angle.
a. If G is complementary to
b. If J is supplementary to
c. If N is vertical to
22.
and
23.
and
angle?
, and
, then
, and
, and
are vertical angles.
are complementary. If
, then
, then
must be
.
must be
.
must be
. Find
.
.
is 18 greater than
, what is the measure of each
and
24.
and
are adjacent angles.
and
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
. What is
?
55
25.
and
angle?
are supplementary angles. If
is 9 less than
, what is the measure of each
and
26. In the diagram below, find all missing angles. Justify with the appropriate angle relationship.
Angle
F
Measure
Justification
ÐFGH
I
J
64°
G
K
ÐIGH
ÐKGH
H
ÐKGJ
27. Given the measures of the following angles, identify the possible angle relationship(s) and explain.
a.
and
Complementary, adjacent angles, their sum is 90 .
b.
and
Congruent angles. Possibly vertical because they share a vertex.
28. For the triangle below, write an equation and solve for x. Then state the measures of
and
.
29. Determine which of these statements is sometimes (not always) true:
a. Supplementary angles add up to 180 .
b. If two lines intersect, each pair of vertical angles are complementary.
c. If the measure of an angle is represented by x, then the measure of its complement is represented by
90 – x.
For the statements that you chose as “sometimes true,” provide one example of when the statement is
true and one example of when the statement is not true. Your examples should be a diagram with the
angle measurements labeled. B is sometimes true. The vertical angles could be 45 , but they could also
measure something different such as 64
SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016
56