1. Objective: To use the Pythagorean Theorem and its converse.
Chapter 7 Lesson 2
Objective: To use the Pythagorean Theorem and its converse.

2. Theorem 7-4  
Pythagorean 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

3. 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.
                                                                                                                                                                                                                                               
3,4,5 5,12,13 8,15,17 7,24,25

4. Example 1: Pythagorean Triples
Find 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.

5. 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?
Not a Pythagorean Triple

6. Example 3: Using Simplest Radical Form
Find the value of x. Leave your answer in simplest radical form.     

7. 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.

8. Example 5: Find the area of the triangle. 102+h2=122 100+h2=144 h2=44
A=(1/2)bh
A=(1/2)(20)(2√11)
A=20√11

9. Example 6: Find the area of the triangle. 72+b2=(√53)2 49+b2=53 b2=4
√53 cm
7 cm
A=(1/2)bh
A=(1/2)(7)(2)
A=7

10. Theorem 7-5: Converse of the Pythagorean Theorem
If the square of the length 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.

11. Example 7: Is this triangle a right triangle? c2=a2+b2 852=132+842
7225=
7225=7225 13 84 85

12. Example 8:
A triangle has sides of lengths 16, 48, and 50. Is the triangle a right triangle?
c2=a2+b2
502=
2500=
2500≠2560
Not a right triangle.

13. If c2>a2+b2, the triangle is obtuse.
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.
If c2>a2+b2, the triangle is obtuse. c a b

14. If c2<a2+b2, the triangle is acute.
Theorem 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.
If c2<a2+b2, the triangle is acute. b a c

15. Example 9:
The lengths of the sides of a triangle are given. Classify each triangle as acute, obtuse, or right.
b.) 12,13,15
152=
225=
225<313
ACUTE
a.) 6,11,14
142=62+112
196=36+121
196>157
OBTUSE

16. Assignment:
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