Guided Notes Converse of Pythagorean Theorem

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Warm-Up Converse to the Pythagorean Theorem

? Lesson Question

Lesson Goals

Use the converse of the Pythagorean theorem.Use the converse of the Pythagorean theorem.

Determine whether a triangle is a

triangle.

Apply the converse of the

real-world scenarios.

theorem to

Words to Know

Write the letter of the definition next to the matching word as you work through the lesson. You may use the glossary to help you.

substitute

deduce

Pythagorean theorem

converse

right triangle

A. to infer; to draw a conclusion

B. to take the place of; to replace

C. a triangle having an interior angle measuring 90 degrees

D. the theorem stating that the sum of the squares of the lengths of the legs in a right triangle is equal to the square of the length of the hypotenuse

E. statement formed by switching the hypothesis and the conclusion of a conditional

WK2

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Warm-Up Converse to the Pythagorean Theorem

Pythagorean Theorem Review

Use the Pythagorean theorem to determine the height of the tree.

• a2 + b2 =

a2 + b2 = c2

a2 + 212 =

a2 + = 1225

−441   −441

a2 =

= 7842a

a = ft

35 ft

21 ft

a

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Instruction

2Slide

Converse to the Pythagorean Theorem

Consider the Pythagorean Theorem

Consider the Pythagorean theorem.

• The sum of the squares of the legs in a right triangle is

equal to the of the length of the hypotenuse.

• If a triangle with sides a, b and c is a triangle, then a2 + b2 = c2.

• What can you deduce about a triangle with side lengths 6, 8, and 10?

a

b

c

Using the Converse of the Pythagorean Theorem

Pythagorean theorem

• If a triangle with sides a,

b and c is a

triangle, then a2 + b2 = c2.

to find

side lengths

of the Pythagorean

theorem

• If a triangle has sides a, b and c such that

+ b2 = c2, then

the triangle is a right triangle.

determine if a triangle is a right triangle

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Instruction Converse to the Pythagorean Theorem

5Slide

Apply the Converse of the Pythagorean Theorem

EXAMPLE

Is a triangle with lengths 15, 20, and 25 a right triangle?

• Apply the converse by the values into the Pythagorean theorem.

a2 + b2 = c2

152 + = 252

225 + 400 =

625 = 625

Since substituting these sides into the formula resulted in a true statement, the side lengths 15, 20, and 25 do form a right triangle.

20

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Instruction Converse to the Pythagorean Theorem

7Slide

Verify the Converse of the Pythagorean Theorem

Verify that 8, 15 and 17 is a Pythagorean triple.

• Does the sum of the squares of the two shorter sides equal the square of the longest side?

a2 + b2 =

+ 152 = 172

64 + = 289

289 = 289

Therefore, this is a right triangle and these numbers are a Pythagorean

.

817

15

225

64

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Instruction Converse to the Pythagorean Theorem

9Slide

12

Using the Converse of the Pythagorean Theorem

EXAMPLE

Does a triangle with side lengths 10, 70 , and 30 form a right triangle?

a2 + b2 = c2

70 10

2 22+

=

70 + 30 = 100

= 100

8 70

30 6

< <

< <

Since these three side lengths satisfy the Pythagorean theorem, I know that they

do form a triangle.

Interpret a Real-World Scenario

REAL-WORLD CONNECTION

A window frame appears to be rectangular. The sides of the window frame are 82 inches and 60 inches, while the length of the diagonal is 102 inches. Does the corner of the window frame form a right triangle for the window to be rectangular?

a2 + b2 = c2

+ 822 = 1022

3,600 + =

10,324 ≠ 10,40460

82?

?

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Instruction Converse to the Pythagorean Theorem

14Slide

Right Triangle Versus Not Right Triangle

EXAMPLE

Right triangle

• A support on a bridge has sides that measure 2.5 m, 6 m and 6.5 m.

a2 + b2 = c2

2.52 + = 6.52

6.25 + 36 =

42.25 = 42.25

Not right triangle

• A kite has sides that measure 16 in., 18 in. and 24 in.

a2 + b2 = c2

+ 162 = 242

324 + = 576

580 576

?

?

6

6.5

18 16

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Summary Converse to the Pythagorean Theorem

Answer

Use this space to write any questions or thoughts about this lesson.

Lesson Question

What is the converse of the Pythagorean theorem and how is it used?

?