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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