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EXPLORE…..
Using one of each color of the Cuisenaire Rods® try to build a right triangle.
1. Use the Cuisenaire Rods® to build a square on each side of the triangle on your paper.
2. With your group, find three different ways of showing that the combined area of the two smaller squares is the same as the largest square. (Rearrange the pink and green rectangles so that they fit ON TOP OF the yellow square.)
4
16 u2
39 u2
4
16 u2
3
9 u2
5
25 u2
Proof
Pythagoras is best known for the Pythagorean Theorem, which relates the side lengths of a right triangle.
The two sides that make up the right angle are called legs.
The side opposite the right angle is the hypotenuse.
leg
leg
hypotenuse
right angle
The Pythagorean Theorem
a
b
c
The Pythagorean Theorem
In a right triangle, if a and b are the measures of the legs and c is the
hypotenuse, then
a2 + b2 = c2.
Note: The hypotenuse, c, is always the longest side.
Baseball Problem
A baseball “diamond” is really a square.
You can use the Pythagorean theorem to find distances around a baseball diamond.
Baseball Problem
The distance between consecutive bases is 90feet. How far does a catcher have to throwthe ball from home plate to second base?
Baseball Problem
To use the Pythagorean theorem to solve for x, find the right angle.
Which side is the hypotenuse?
Which sides are the legs?
Now use: aa22 + b + b22 = c = c22
Baseball ProblemSolution
• The hypotenuse is the distance from home to second, or side x in the picture.
• The legs are from home to first and from first to second.
• Solution: x2 = 902 + 902 =
16,200 x = 127.28 ft
Check It Out: Example 1A
5
12
c
13 = c
Pythagorean TheoremSubstitute for a and b.
a2 + b2 = c2
52 + 122 = c2
25 + 144 = c2
169 = cSimplify powers. Solve for c; c = c2.
Find the length of the hypotenuse to the nearest hundredth.
10
b
26
Finding c (HYPO) – square numbers, ADD, then take the square root!
Finding a or b (LEG) – square numbers, SUBTRACT,
then take the square root!
STEP One: Identify the hypo and legs!
COPY THIS INTO YOUR
NOTES!
10
b
26
= 24
(a)
(c)
222 cba 222 2610 b
676100 2 b1006762 b
5762 b24b
1m
8m
c
b=
a=
c²=a²+ b²
c²=1²+ 8²
c²=1 + 64
c²=65
?
Using Pythagoras’ Theorem
Example 1
c
12cm
9cm
a
ba²+ b²= c2
12²+ 9²= c²
144 + 81 =c²
c²= 225
c = √225= 15cm
c
6m4m
s
ab
a²+ b²= c2
4²+ 6²= c²
16 + 36 = c²
c²= 52
c = √52
=7.2m (1 d.p.)
Example 2
7m
5m
hc
a
b
a²+ b²= c²
a²+ 5²= 7²
a² + 25 = 49?
Finding the shorter side
a² + 25= 49
We need to get a² on its own.Remember, change side, change sign!
Finding the shorter side
- 25
a²= 49 - 25 =a²= 24
a = √24 = 4.9 m (1 d.p.)
169 = w² + 36
c
w
6m
13m
a
b
c²= a²+ b²
13²= a²+ 6²
169 – 36 = a²
a = √133 = 11.5m
(1 d.p.)
a²= 133
Example 1
169 = a² + 36
Change side,
change sign!
c
b c²= a²+ b²
11²= 9²+ b²
121 = 81 + b²
121 – 81 = b²
b = √40 = 6.3cm
(1 d.p.)
b²= 40
a9cm
P
11cm
R
Q
Example 2
81
Change side,
change sign!
You can use The Pythagorean Theorem to solve many kinds of problems.
Suppose you drive directly west for 48 miles,
48
Then turn south and drive for 36 miles.
48
36
How far are you from where you started?
48
36?
482
Using The Pythagorean Theorem,
48
36c
362+ = c2
Why? Can you see that we have a right triangle?
48
36c
482 362+ = c2
Which side is the hypotenuse? Which sides are the legs?
48
36c
482 362+ = c2
22 3648
Then all we need to do is calculate:
12962304
3600 2c
And you end up 60 miles from where you started.
48
3660
So, since c2 is 3600, c is 60.
1. The measures of three sides of a triangle are given below. Determine whether each
triangle is a right triangle. 7 , 3, and 8
Which side is the biggest?
8 This must be the hypotenuse (c).
Plug your information into the Pythagorean Theorem. It doesn’t matter which number
is a or b.
9 + 49 = 64 ?58 = 64 ?
This is NOT true, it is NOT a right triangle.
Sides: 7 , 3, and 832 + 72 = 82
Determine whether the triangle is a right triangle given the sides 9, 15, and 12
1. Yes
2. No
3. Purple
92 + 122 = 152
81 + 144 = 225
Converse of the Pythagorean Theorem
Theorem 8-2: Converse of the Pythagorean Theorem – If the square of the lengths of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
If a2+b2=c2, then the triangle is a right triangle.
If , complete each statement.
Applying the Converse of the Pythagorean Theorem
Are the following triangles right triangles? EXPLAIN
8584
13
21
20
28
842 + 132 = 852
7056 + 169 = 7225
7225 = 7225
YES!
212 + 202 = 282
441 + 400 = 784
841 = 784
NO!
Classifying Triangles
Determine if the Triangle is a Right Triangle
Given a triangle with sides 6, 11, and 14 how can you classify the triangle?
Given a triangle with sides 7, 8, and 9 how can you classify the triangle?
62 + 112 = 142
36 + 121 = 196
157 = 196
NO!
72 + 82 = 92
49 + 64 = 81
113 = 81
NO!
Application
The Parks Department rents paddle boats at docks near each entrance to the park. To the nearest meter how far is it to paddle from one dock to the other?
Use the Pythagorean Theorem to determine if the following are right triangles.
1. 11 cm, 60 cm, 61 cm
2. 5 ft, 12 ft, 15 ft
3. 17 in, 9 in, 15 in
4. 52 cm, 20 cm, 48 cm