Park Forest Math Team
Meet #3
Self-study Packet
Problem Categories for this Meet (in addition to topics of earlier meets):
1.
2.
3.
4.
5.
Mystery: Problem solving
Geometry: Properties of Polygons, Pythagorean Theorem
Number Theory: Bases, Scientific Notation
Arithmetic: Integral Powers (positive, negative, and zero), roots up to the sixth
Algebra: Absolute Value, Inequalities in one variable including interpreting line graphs
Important Information you need to know about GEOMETRY…
Properties of Polygons, Pythagorean Theorem
Formulas for Polygons where n means the number of sides:
• Exterior Angle Measurement of a Regular Polygon: 360÷n
• Sum of Interior Angles: 180(n – 2)
• Interior Angle Measurement of a regular polygon:
• An interior angle and an exterior angle of a regular polygon
always add up to 180°
Interior
angle
Exterior
angle
Diagonals of a Polygon where n stands for the number of vertices (which is equal
to the number of sides):
•
• A diagonal is a segment that connects one vertex of a polygon to
another vertex that is not directly next to it. The dashed lines represent
some of the diagonals of this pentagon.
Pythagorean Theorem
• a2 + b2 = c2
• a and b are the legs of the triangle and c is the hypotenuse (the side
opposite the right angle)
c
a
b
•
Common Right triangles are ones with sides 3, 4, 5, with sides 5, 12, 13,
with sides 7, 24, 25, and multiples thereof—Memorize these!
Category 2
Geometry
Meet #3 - January, 2015
Figures are not necessarily drawn to scale.
1) Angle ADC is a right angle. AB = 4 cm and BC = 9 cm. DB is
perpendicular to AC. How many cm long is DB ?
D
A
C
B
2) DEHJ is a square with an area of 64 square meters. Diagonal
HF = 17 meters. How many square meters are in rectangle DFGJ ?
D
E
F
J
H
G
3) Polygon PENTAGONAL is a pentagram (star) consisting of a regular
pentagon with five isosceles triangles attached at its five edges. How
many degrees are in one of the exterior
A
N
angles (for example, angle PEN) ?
Answers
T
G
E
1)
L
2)
N
P
3)
A
O
Solutions to Category 2
Geometry
Meet #3 - January, 2015
1) A student who knows the Pythagorean Theorem
should also know that, at its foundation, is the notion
of similar triangles. In this diagram are three similar
triangles. Using triangles DAB and DBC, we can say
that ratios of corresponding sides are proportional:
. So,
and cross products are
Answers
1)
6
2) 184
3) 108
equal, so
and DB = 6. A few students
may recognize this diagram as representing this
theorem: "The altitude to the hypotenuse of a right triangle is the
geometric mean (or mean proportional) to the two segments of the
hypotenuse into which it is divided."
2) One side of square DEHJ is 8 meters because its area is 64
square meters. For one of the right triangles of rectangle EFGH,
using the Pythagorean Theorem,
.
So,
, and
, so
,
and EF = 15. So, rectangle DFGJ now measures 8 by (15 + 8),
or 8 by 23, so its area is (8)(23), or 184 square meters.
3) Each interior angle of the regular pentagon measures (3)(180)/5,
or 108 degrees. Any one of the exterior angles of the pentagram is
vertical to one of these interior 108 degree angles and, therefore, is
equal to 108 degrees.
Category 2 Geometry Meet #3, January 2013 1. Mia drew regular hexagons on each side of a pentagon. If she draws all the diagonals in all six shapes, how many diagonals will she have to draw? Note: A diagonal is a line segment that connects two vertices that are not already connected by a side. 2. Right triangle ABC below has legs of length 7 units and 11 units. How many square units are there in the area of square ACDE which is constructed on the hypotenuse of this triangle? 3. A small rectangular box has sides of lengths 6 cm, 6 cm, and 7 cm. How many centimeters are there in the space diagonal of the box? Note: A space diagonal is a line that connect two opposite vertices of the box and goes through the interior space of the box. Answers 1. ___________ diagonals 2. ____________ sq. units 3. __________________ cm Answers Solutions to Category 2 Geometry Meet #3, January 2013 1. 50 diagonals 2. 170 sq. units 3. 11 cm 1. Three diagonals can be drawn from each of the six vertices on a hexagon, but this would count each diagonal at both ends. So there are 3 × 6 ÷ 2 = 9 diagonals in each hexagon. Similarly, there are 2 × 5 ÷ 2 = 5 diagonals in the pentagon. Mia will have to draw 5 × 9 + 5 = 50 diagonals. 2. According to the Pythagorean theorem, the sum of the areas of the squares on the legs of a right triangle is equal to the area of the square on the hypotenuse. If we were to construct a square on leg AB, it would have an area of 11 × 11 = 121 square units. A square on leg BC would have an area of 7 × 7 = 49 square units. Their sum is 121 + 49 = 170 square units and this is the area of square ACDE. 3. We can calculate the length of the space diagonal of the box by using the Pythagorean theorem twice. First we can find the length of the diagonal of the bottom face, which is 6 2 + 6 2 = 72 = 6 2 cm. Then we find the length of the space diagonal using this length and the height of ( )
2
the box as follows: 72 + 6 2 = 49 + 72 = 121 = 11 cm. €
Alternatively, we can use a 3-­‐dimensional version of the Pythagorean theorem as follows: 6 2 + 6 2 + 72 = 36 + 36 + 49 = 121 = 11 cm. €
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Category 2
Geometry
Meet #3, January 2009
1.
How many degrees are in the sum of the interior angles of a convex
decagon?
2.
Let the number of diagonals in a regular octagon be !, and the number of
diagonals in a regular hexagon be ". What is the value of ! # " ?
3.
Quadrilateral ABCD has right angles at A and C. The lengths of CD, BC,
and AB are 7 cm, 11cm, and 1cm respectively. How many centimeters long
is AD?
C
B
A
D
Answers
1. _______________
2. _______________
3. _______________
Solutions to Category 2
Geometry
Meet #3, January 2009
Answers
1. 1440
2.
11
3.
13
1. For an !-sided convex polygon, the expression
"#$%! & '( will give the number of degrees in the sum of the
interior angles of the polygon. So a decagon %! ) "$( would
have "#$%"$ & '( ) "#$%#( ) "**$ degrees.
+%+,-(
2. For an !-sided convex polygon, the expression
will
.
give the number of diagonals in the polygon. So an octagon has
/%0(
1%-(
) '$ diagonals and a hexagon has
) 2 diagonals. So
.
.
3 ) '$ and 4 ) 2. Therefore 3 & 4 ) '$ & 2 ) "".
3. First draw in segment 56. Since 56 is the hypotenuse of triangle 576, we
can use the Pythagorean Theorem to find the length of drawn in segment 56.
56. ) 8. 9 "". ) *2 9 "'" ) "8$. However BD is also the hypotenuse of
triangle :56 and therefore 56. ) ". 9 :6. ) " 9 :6. ) "8$. So :6. ) ";2
and :6< ) <"=.
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C
B
7
1
A
?
D
Category 2
Geometry
Meet #3, January 2007
1. How many degrees are in the measure of an exterior angle of a regular
decagon?
exterior angle
2. A Pythagorean Triple is a set of three natural numbers that satisfy the
Pythagorean Theorem a 2 + b 2 = c 2 , where a and b are legs on a right triangle and c
is the hypotenuse. One way to find Pythagorean Triples is with the three
equations a = m 2 − n 2 , b = 2mn , and c = m 2 + n 2 , where the values of m and n are
natural numbers with m > n. How many units are in the perimeter of the right
triangle that it is produced when m = 7 and n = 4?
3. An isosceles triangle has sides measuring 34 units, 34 units, and 32 units. If the
32-unit side is considered the base, how many units are in the height (or the
altitude) of this triangle?
Answers
1. _______________
2. _______________
3. _______________
34 units
34 units
32 units
www.imlem.org
Solutions to Category 2
Geometry
Meet #3, January 2007
Answers
1. 36
2. 154
1. Imagine taking a walk counter-clockwise around the
regular decagon. You will make ten left-hand turns of
the same measure. When you get back to where you
started, you will have turned a total of 360 degrees.
Therefore, each exterior angle must be 360 ÷ 10 = 36
degrees.
3. 30
2. Using m = 7 and n = 4, we find that
a = 7 2 − 4 2 = 49 −16 = 33 , b = 2 ⋅ 7 ⋅ 4 = 56 , and
c = 7 2 + 4 2 = 49 + 16 = 65 . The perimeter of a right
triangle with sides of 33, 56, and 65 units is 33 + 56 + 65
= 154 units.
3. First we draw the height we wish to find. An altitude
line is always perpendicular to the base. On an isosceles
triangle this line also meets the base at the midpoint.
Now we can use the Pythagorean Theorem to find the
missing leg of a right triangle with an hypotenuse of 34
units and a leg of 16 units. 342 = 1156 and 162 = 256.
1156 – 256 = 900 and 900 = 30 , so the height of the
isosceles triangle must be 30 units.
34 units
34 units
16 units and 16 units
www.imlem.org
Category 2
Geometry
Meet #3, January 2005
1. A certain polygon has twice as many diagonals as sides. How many sides are
there on this polygon?
Note: A diagonal in a polygon is any line segment that connects two vertices and
is not a side.
2. The figure below shows the design of a raft which floats in the middle of a
lake. The raft is made of a number of square sections that are linked together.
If the perimeter of the raft is 220 feet, how many square feet are in the area of the
raft?
3. In the figure below, triangles ABC, ACD, and ADE are right triangles, and
sides AB, BC, CD, and DE have the same measure. If the measure of side AB is 2
centimeters, how many centimeters are there in the measure of side AE?
E
D
C
Answers
1. _______________
2. _______________
3. _______________
B
www.Imlem.org
A
Solutions to Category 2
Geometry
Meet #3, January 2005
Average team got 18.63 points, or 1.55 questions correct
1. The table below shows the number of diagonals for
several polygons. The heptagon with 7 sides has 14
diagonals, which is twice the number of sides.
Answers
1. 7
Polygon
Triangle
Square
Pentagon
Hexagon
Heptagon
Octagon
2. 625
3. 4
Sides
3
4
5
6
7
8
Diagonals
0
2
5
9
14
20
2. There are 44 side lengths of the square sections in the
perimeter of the raft, so each square must have a side
length of 220 feet ÷ 44 = 5 feet. The area of each square
is thus 5 feet × 5 feet = 25 square feet.
The raft is made of 25 sections, so the area of the raft is
25 × 25 square feet = 625 square feet.
E
D
C
3. We will have to use the Pythagorean Theorem three
times to calculate the length of each hypotenuse. Let the
length of AC = x, the length of AD = y, and the length of
AE = z. Then we find x as follows:
x2 = 22 + 22  x2 = 8  x = 8  x = 2 2
We find y as follows:
y2 = 22 +
B
A
( 8)
2
 y 2 = 12  y = 12  y = 2 3
Finally, we find z as follows:
z 2 = 22 +
( 12 )
2
 z 2 = 16  z = 4 .
AE is 4 centimeters.
www.Imlem.org
So the length of