Page 171 - Math Course 2 (Book 1)

The Pythagorean Theorem and Its Converse

2 2 2 BC = 52 , AC = 52
104 )
52 ) ( ) =
Let’s Begin ( 52 ( AB = 104
52 + 52 = 104 Simplify.

Verify a Triangle is a Right 104 = 104 Add.
Triangle
Since the sum of the squares of
two sides equals the square of
Example Answer the longest side, ΔABC is a right
triangle.

COORDINATE GEOMETRY
Verify that ΔABC is a right triangle. Pythagorean Triples
y

x Examples
0
A(–9, –3) B
(1, –1)
Determine whether 9, 12, and 15 are the sides of
a right triangle. Then state whether they form a
Pythagorean triple.

Since the measure of the longest side is 15, 15
C(–3, –7) must be c. Let a and b be 9 and 12.

2
2
Use the Distance Formula to determine the lengths a + b = c 2 Pythagorean Theorem
of the sides. 2 2 ? 2
2
AB = [–1–(–9)] + [–1–(–3)] 2 x = –9, y = –3, 9 + 12 = 15 a = 9, b = 12, c = 15
1
1
x = 1 , y = –1 ?
2 2 81 + 144 = 225 Simplify.
2
= 10 + 2 2 Subtract.
225 = 225 Add.
= 104 Simplify.
These segments form the sides
2
BC = (–3–1) + [–7–(–1)] 2 x = 1, y = –1, of a right triangle since they
1
1
x = –3, y = –7 Answer satisfy the Pythagorean Theorem.
2
2
The measures are whole numbers
2
= (–4) + (–6) 2 Subtract. and form a Pythagorean triple.
= 52 Simplify.
Determine whether 4 , 4, and 8 are the sides
3
of a right triangle. Then state whether they form a
2
AC = [(–3–(–9)] + [–7–(–3)] 2 x = –9, y = –3, Pythagorean triple.
1
1
x = –3, y = –7
2 2
2
2
2
= 6 + (–4) 2 Subtract. a + b = c 2 Pythagorean Theorem
2 ?
2
= 52 Simplify. (4 3) + (4) = 8 2 a = 4 3, b = 4, c = 8
?
48 + 16 = 64 Simplify.
By the converse of the Pythagorean Theorem, if the
sum of the squares of the measures of two sides of 64 = 64 Add.
a triangle equals the square of the measure of the
longest side, then the triangle is a right triangle. The segments form the sides of
Answer a right triangle, but the measures
2
2
(BC) + (AC) = (AB) 2 Converse of the do not form a Pythagorean triple.
Pythagorean Theorem.
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