chapter 7 lesson 2
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Chapter 7 Lesson 2. Objective: To use the Pythagorean Theorem and its converse. 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. a 2 + b 2 = c 2. - PowerPoint PPT PresentationTRANSCRIPT
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 5,12,125,12,12 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:
Page 361 – 363
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