You will need
10.7
• a calculator
Solve Problems Using
Logical Reasoning
GOAL
Use logical reasoning to solve problems.
Learn about the Math
1st Ave.
2nd Ave.
3rd Ave.
4th Ave.
5th Ave.
A Street
n
ee t
Qu ree
St
On this city map, the blue lines show major
roads. Notice the following:
• One angle measure is given.
• All lettered streets are parallel to each other.
• All numbered avenues are parallel to each
other.
• Queen Street is perpendicular to King Street.
• 1st Avenue is perpendicular to A Street.
• Queen Street crosses 3rd Avenue halfway
between C and D Streets.
• King Street crosses D Street halfway between
1st and 2nd Avenues.
library school
B Street
ng
K i eet
r
St
C Street
pool
mall
Problem: A car starts at one of the labelled red
dots (school, pool, library, or mall). It
travels less than 1 km and crosses a
street. After travelling 1.8 km, it turns
right at an angle of 56°. Then, 4 km
later, it turns right, travels 5 km, turns
left, and travels 1.2 km to its
destination.
124˚
D Street
E Street
Scale: 1 unit 1 km
? Where did the car start and end?
1 Understand the Problem
Annika understands that she needs to determine the path of the car
from one of the red dots and end up at a different red dot after
following the directions.
2 Make a Plan
Annika plans to eliminate some starting and ending locations by
using only the first and last pieces of information in the problem.
358 Chapter 10
NEL
3 Carry Out the Plan
First, Annika determines which of the four locations are less than
1 km from an intersection.
“If I draw a line from the school perpendicular to 3rd Avenue, the
distance is 1 km. The road to the intersection is the hypotenuse of a
right triangle. So, it must be more than 1 km from the intersection.
school
1 km
“I can use the same logic to show that the library is more than 1 km
from the closest intersection.
“The only possible starting points are the mall and the pool.”
“I’ll look for places where the car can travel 5 km between
intersections. The possibilities are
library
1 km
Next, Annika determines the possible ending locations.
• along 2nd Avenue from B Street to King Street
• along 4th Avenue from D Street to King Street
• along C Street from 1st Avenue to Queen Street
“I followed all these routes. I found that only when travelling along
C Street can you turn left and go a little over 1 km to a red dot. The
ending location must be the library.
“I’ll work backward from the library to find the starting location.
A distance of 1.2 km takes the car back to C Street. Turning right
(opposite of turning left because I’m going backward) takes the car
onto C Street. Then 5 km takes the car to 1st Avenue. Turning left and
travelling 4 km takes the car to King Street. Turning left at a 56° angle
onto King Street and travelling 1.8 km takes the car to the mall.
“The car started at the mall and ended at the library.”
4 Look Back
Annika followed the directions starting at the mall. The car ended at
the library.
Reflecting
1. Why is it reasonable to assume that the library is 1.2 km from the
intersection of C Street and Queen Street?
2. Looking back, Annika said that the key piece of information she
used was the car travelling 5 km between two intersections. Why did
this help her eliminate so many possibilities?
3. Annika could have calculated all the angles and distances on the map
and tested all the routes. How did logical reasoning reduce her work?
NEL
Angles and Triangles
359
Work with the Math
Example: Eliminating possibilities by using a table
Three people are in a room. Use the following clues to figure out their names
and occupations.
Clue 1: Their first names are Anna, Peter, and Roger.
Clue 2: Two of the last names are Pavan and Attwell.
Clue 3: One person’s first and last names begin with the same letter as her or
his occupation.
Clue 4: One person is an architect.
Clue 5: The person whose last name is Davis is a pilot.
Clue 6: Roger is not a singer.
Ken’s Solution
1
Understand the Problem
I need to figure out which combination works for each person.
2
Make a Plan
I’ll make a chart of all the combinations and mark the combinations that don’t work.
3
Carry Out the Plan
From clues 1 to 4, either Anna Attwell could be the architect or Peter Pavan could be
the pilot. But from clue 5, Davis is the pilot. So, Anna Attwell must be the architect.
(From this, I can complete the first row and the third and fifth columns in my chart.)
From clue 6, Roger is not the singer. So, he must be Roger Davis, the pilot. From this,
I can complete the rest of the chart. Peter Pavan is the singer.
4
Pavan
Attwell
Davis
architect
pilot
Anna
X
Peter
yes
Roger
X
singer
yes
X
yes
X
X
X
X
X
X
yes
X
yes
X
yes
no
Look Back
I checked to make sure that my answer works with the clues that were given.
A
Checking
B
4. The hypotenuse in a right scalene triangle
is 7.0 cm. What must be the length of one
of the legs?
A. 8.0 cm
C. 5.5 cm
B. 4.95 cm
D. 7.0 cm
360 Chapter 10
Practising
5. Five students are walking in single file.
Devon is not second. Wendy is just behind
Devon. Hannah is just behind Kathryn,
who is not third. Gregory is neither first
nor last. Who is third? How do you know?
NEL
6. Suppose that you know the measure of the
angle with a check mark. Can you
determine the measures of all the angles
marked with dots? Why or why not?
A
8. Draw three triangles. Each triangle should
fit at least two of the following criteria.
Explain your thinking.
• The side lengths you can choose from
are 3 cm, 4 cm, 4 cm, 4 cm, 5 cm, 5 cm,
5 cm, 5 cm, and 7 cm.
• One triangle is acute.
• One triangle is obtuse.
• One triangle is a right triangle.
• One triangle is an equilateral triangle.
• One triangle is isosceles, but not
equilateral.
• One triangle is scalene.
C
E
F
G
H
D
B
7. How many bananas weigh the same as one
cantaloupe?
2 cantaloupe 1 grapefruit 1 banana
9. All the marked sides of the polygons are
4 cm. Explain how to order the polygons
from least to greatest area.
1 grapefruit 5 bananas
60˚
1
2
3
4
SQUARING NUMBERS
THAT END IN 5
You can square a two-digit number that
ends in 5 by calculating the area of a
square. To calculate 252, for example,
think of the area of a 25-by-25 square.
5
20
20 20 400
5 20 100
To square 25, first square 20, then
double 100 (which is 5 20), and then
square 5. Add the four areas to get the
total area.
1. Use this mental math strategy to
calculate each square.
a) 152
b) 452
c) 752
20
5
5 20 100
55
25
2. Use your answers to question 1 to calculate each square.
a) 1.52
b) 3.52
c) 5.52
d) 7.52
e) 9.52
NEL
20 5
20 5
400
100
100
25
625
202
5 20
5 20
55
f) 8.52
Angles and Triangles
361