Chapter 7 Lesson 2Chapter 7 Lesson 2

Objective: ToObjective: To use the Pythagorean Theorem and its

converse.

Theorem 7-4  Theorem 7-4  Pythagorean TheoremPythagorean Theorem

In a right triangle, the sum of the squares of the lengths of the legs is

equal to the square of the length of the hypotenuse. a2 + b2 = c2

A Pythagorean triple is a set of nonzero whole numbers a, b, and c that satisfy the equation a2 + b2 = c2. Here are

some common Pythagorean triples.

                                                                                                                                                                                             

                                                  

3,4,53,4,5 5,12,135,12,13 8,15,178,15,17 7,24,257,24,25

Example 1: Pythagorean TriplesFind the length of the hypotenuse of ∆ABC. Do the lengths of the sides of ∆ABC form a

Pythagorean triple?

                                                                                                                                            

        

The lengths of the sides, 20, 21, and 29, form a Pythagorean triple.

Example 2: Pythagorean Triple

A right triangle has a hypotenuse of length 25 and a leg of length 10. Find the length of the other leg. Do the lengths of the sides form a Pythagorean triple?

222 cba 222 2510 a

6251002 a5252 a525a2125a

215aNot a Pythagorean Triple

Example 3: Using Simplest Radical Form

Find the value of x. Leave your answer in simplest radical form.     

Example 4: Using Simplest Radical Form

The hypotenuse of a right triangle has length 12. One leg has length 6. Find the length of the other leg. Leave the answer in simplest radical form. 222 cba

222 126 a

1082 a108a336 a

36a144362 a

Example 5:Example 5:Find the area of the triangle.

h

20 m

12 m12 mh

10 m

12 m

102+h2=122

100+h2=144

h2=44

h=√44

h=2√11

A=(1/2)bh

A=(1/2)(20)(2√11)

A=20√11

Example 6:Example 6:Find the area of the triangle.Find the area of the triangle.

√53 cm

7 cm72+b2=(√53)2

49+b2=53

b2=4

b=2

A=(1/2)bhA=(1/2)(7)(2)A=7

Theorem 7-5:Theorem 7-5:Converse of the Pythagorean Converse of the Pythagorean

TheoremTheorem

If the square of the length If the square of the length of one side of a triangle is of one side of a triangle is

equal to the sum of the equal to the sum of the squares of the lengths of squares of the lengths of the other two sides, then the other two sides, then

the triangle is a right the triangle is a right triangle.triangle.

Example 7:Example 7:Is this triangle a right triangle?

c2=a2+b2

852=132+842

7225=169+7056

7225=7225

13

84

85

Example 8:Example 8:A triangle has sides of lengths 16, 48, A triangle has sides of lengths 16, 48, and 50. Is the triangle a right triangle?and 50. Is the triangle a right triangle?

c2=a2+b2

502=162+482

2500=256+2304

2500≠2560

Not a right Not a right triangle.triangle.

Theorem 7-6:Theorem 7-6: If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, the triangle is obtuse.

a

b

c

If c2>a2+b2, the triangle is obtuse.

If c2<a2+b2, the triangle is acute.

Theorem 7-7Theorem 7-7::If the squares of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, the triangle is acute.

ab

c

Example 9:Example 9:The lengths of the sides of a triangle are given. Classify each triangle as acute, obtuse, or right.

a.) 6,11,14142=62+112

196=36+121

196>157

OBTUSE

b.) 12,13,15

152=132+122

225=169+144

225<313

ACUTE

Assignment:Assignment:

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