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Pythagorean Theorem

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Square C

Square B

Square A

AC

B

Vocabulary:

Hypotenuse – the side across from the right angle, it will be the longest side

Legs – are the sides adjacent to the right angle

His theorem states: 222 cba

In any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the

hypotenuse.

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Proof of Pythagorean’s Theorem:

1. How do the areas of the two squares compare?

2. Cut out 4 of the right triangles and the two smaller squares (a and b) on page 5.

How do the legs of the triangle compare to the sides of the two squares? Label the area of each square.

3. Take these 6 pieces; arrange them in a manner to completely cover the square on the left above. (think:

puzzle)

4. Cut out the remaining 4 right triangles and the remaining square. How does the hypotenuse compare to

the side of the square? Label the area of the square.

5. Take these 5 pieces; arrange them in a manner to completely cover the square on the right above. (think:

puzzle)

6. Explain how the above diagram proves the Pythagorean Theorem: 222 cba

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Converse of Pythagorean Theorem:

If a triangle has sides of lengths a, b and c, and 222 cba , then the triangle is a right triangle with

hypotenuse of length c.

You can use the converse of the Pythagorean Theorem to determine whether a triangle is a right triangle.

Since the Pythagorean Theorem and its converse are always true, you can also determine whether a triangle is

not a right triangle.

Determine if the given sides are sides of a right triangle.

Ex. 1: 5 in, 12 in, 13 in Ex. 2: 7m, 9m, 12m

Ex. 3: 7.6 cm, 11.5 cm, 8.4 cm Ex. 4: 10,200,10

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HW # 5 - Pythagorean Theorem & It’s Converse

Determine if the given sides are sides of a right triangle. Show your work.

1. 1 in, 2 in, 3 in 2. 50,5,5

3. 8, 17, 15 4. 13, 17, 15

5. 3 4

, ,15 5

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Proof of Pythagorean’s Theorem:

1. Cut out the pieces of Figure 1 on page 3.

2. Rearrange the pieces to cover the two dotted squares on page 3.

3. Explain how the diagram proves the Pythagorean Theorem: 222 cba

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Pythagorean Theorem – Missing Hypotenuse

Find the missing hypotenuse.

1: 2:

3: Find the exact length of the hypotenuse.

4: Round the answers to the nearest tenth.

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5. A rectangular field with a length of 80 yards and a width of 60 yards; what is the length of the diagonal?

6. A string is tied to the top of a 5 foot flagpole and pulled taut to the ground. The base of the flagpole is 9

feet away from where the string is tied to the ground. What is the length of the string to the nearest

hundredth of a foot?

7. A 26 foot long ramp is used to reach 10 ft up a wall. The ramp covers a horizontal distance of 24 ft. To

make the ramp less steep, the horizontal distance is doubled. To reach the same 10 ft height, how long will

the ramp need to be, rounded to the nearest hundredth of a foot?

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Homework

Find the missing hypotenuse of each right .

1. Find the hypotenuse.

2. Find the exact length of the hypotenuse.

3. Find the hypotenuse. (Hint: give the calculator answer.)

a = 1

b = 3

find c

4. Find the length of the hypotenuse to the nearest hundredth.

12

5

c

x

1

10

5. A rectangular soccer field with a length of 100 yards and a width of 60 yards; what is the length of the

diagonal? Round to the nearest tenth.

6. A cable supports a 28 foot utility pole. The cable is fastened to the ground at a point 21 feet from the base

of the pole. Find the length of the cable.

Is the whose sides have the given lengths a right ?

7. 3 4

, ,15 5

8. 16, 9, 25

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Pythagorean Theorem – Missing Leg

Find the missing leg, show all work.

1: Find the missing leg.

2: Find the exact length of the missing leg.

3: Find the missing leg to the nearest tenth.

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4: Find the missing leg.

5: Find the missing leg to the nearest hundredth.

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6. A ladder 12 feet in length leans against a wall. The base of the ladder is how far away from the base of

the wall if the top of the ladder reaches 10 feet up the wall? Round your answer to the nearest tenth.

7. The freight entrance to a store is 3 feet above ground level. An access ramp to the entrance is 10 feet

long. What is the distance from the bottom of the ramp to the base of the building? Round to the nearest

hundredth.

8. If the hypotenuse of a right triangle measures 15 cm and one of its legs is 12 cm, what is the

measurement of the missing leg?

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Practice

Find missing side of each right .

1. Find the missing length.

2. Find the missing length.

3. Find the exact length of the missing side.

a = 16

c = 20

find b

4. Find the missing length to the nearest tenth.

4

5 a

36 24

x

15

5. Find the exact length of the missing side.

6. A ladder 6 meters in length leans against a wall. The base of the ladder is 2 meters away from the base

of the wall. How far up the wall does the ladder reach to the nearest tenth?

7. If the hypotenuse of a right triangle measures 15 cm and one of its legs is 9 cm, what is the measurement

of the missing leg?

Is the whose sides have the given lengths a right ?

8. 2,3,1 9. 26m, 10m, 24m

10. Find the hypotenuse of a right triangle given leg measurements of 8cm and 15cm.

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Applications of Pythagorean Theorem

Example 1 - Finding the Distance between 2 points on a Coordinate Plane:

Find the exact length of AB .

Try It!

Find the exact length of AB .

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Example 2 – Finding Triangle Lengths in the Coordinate Plane

Randy plots the points in the following graph.

Find the lengths of AB , BC , and AC .

What kind of triangle is ABC ? Explain.

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3. If the base of an isosceles triangle is 12 inches and the altitude is 8 inches, what is the perimeter of the

triangle?

4. Often builders want to know the length of wire, cable, or lumber needed to go from one point to another

without having to actually measure the distance beforehand. If the lengths are parts of right triangles, the

Pythagorean formula can be used to find the answer. The next example illustrates such a situation.

The figure shows a sketch of a rectangular storage area that is part of a warehouse. A chute is going to be

built from point A (the back right corner) to point B (the front left corner). The dimensions of the

rectangular area are known: 30ft long, 20ft wide, and 10 ft high. What is the length of the chute – distance

AB?

A

D

B C

10ft

20ft

30ft

AB = ?

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Practice

1. Is XYZ a right triangle? Explain.

2. Two college roommates, Harry and Niall, leave college at the same time. Harry travels south at 30 miles

per hour and Niall travels west at 40 miles per hour. How far apart are they at the end of one hour and

thirty minutes?

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3. In an isosceles triangle, each of the equal sides is 26 centimeters long and the base is 20 centimeters

long. Find the area. (Hint: first draw the altitude to the base.)

4. If a rectangle has a diagonal of 20 feet and a side of 12 feet, find the perimeter and area of the

rectangle.

5. You know that two sides of a right triangle measure 10in and 8in.

a) Why is this not enough information to be sure of finding the length of the third side?

b) Give two different possible values for the length of the third side. Explain how you found your

answers.

6. Find all possible values of x.

D F

E

m

x

x + 7

x + 9

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Mixed Review

Practice: Draw pictures; find the exact length of the missing sides.

1. ?,mi24,mi7 cba 2. ft51?,,ft3 cba

Determine if the given sides are sides of a right triangle.

3. 1 in, 2 in, 3 in 4. 2,1,1

5. Kendrick wants to build a slide for his son in the backyard. He buys a slide that is 8 feet long. The height

of the stairs is 5 feet. Find the distance from the bottom of the stairs to the base of the slide. Show your work,

round your answer to the nearest tenth.

6. Mrs. Hanson uses a wheelchair, her husband decides to build a ramp to make it easier for her to enter

and leave the house. Find the length of the ramp. Show your work.

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7. Find the value of x in the following diagram. Round your answer to the nearest tenth.

8. Often builders want to know the length of wire, cable, or lumber needed to go from one point to another

without having to actually measure the distance beforehand. If the lengths are parts of right triangles, the

Pythagorean formula can be used to find the answer. The next example illustrates such a situation.

The figure shows a sketch of a rectangular storage area that is part of a warehouse. A chute is going to

be built from point A (the back right corner) to point B (the front left corner). The dimensions of the

rectangular area are known: 30ft long, 20ft wide, and 10 ft high. What is the length of the chute –

distance AB?

A

D

B C

9. Mike bought the cone-shaped candle shown. What is the volume of the candle? Round your answer to

the nearest tenth.

10ft

20ft

30ft

AB = ?

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____ 10. Which of the following sides lengths are not sides of a right triangle?

(a) 17,15,8 (c) 13,12,5

(b) 7,6,5 (d) 26,24,10

Determine if the given sides are sides of a right triangle.

11. 2,3,1

12. A straw that is 12cm long leans against the inside of a glass. 3cm of the straw extends beyond the glass.

The height of the glass is 7.5cm. Find the radius of the glass to the nearest tenth.

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1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

13. Points 3,4 A and 4,4B are plotted on a coordinate plane.

Find the exact distance between points A and B.

14. Find the value of x in the following diagram. Round your answer to the nearest tenth.