5-3A The Pythagorean Theorem

You used the Pythagorean Theorem to develop the Distance Formula.

• Use the Pythagorean Theorem.

• Use the Converse of the Pythagorean Theorem.

Pythagorean TheoremThe Pythagorean Theorem is used to calculate

the length of any side of a right triangle when the lengths of the other two sides are known.

Which side is the hypotenuse?

Which sides are the legs?a

b

c

Pythagorean TheoremIn a right triangle, the square of the length

of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

a2 + b2 = c2

b

ac

p. 547

p. 547

Find the length of the third side of the right triangle ∆STU.

ST=3, TU = 4

ST2 + TU2 = SU2

32 + 42 = SU2

9 + 16 = SU2

25 = SU2

5 = SU

S

T U

A. Find x.

The side opposite the right angle is the hypotenuse, so c = x.

a2 + b2 = c2 Pythagorean Theorem

42 + 72 = c2 a = 4 and b = 765 = c2 Simplify.

Take the positive square rootof each side.

Answer:

B. Find x.

The hypotenuse is 12, so c = 12.

a2 + b2 = c2 Pythagorean Theorem

x2 + 82 = 122 b = 8 and c = 12

Take the positive squareroot of each side andsimplify.

x2 + 64 = 144 Simplify.

x2 = 80 Subtract 64 from each side.

Answer:

Find the length of the third side of the right triangle ∆STU.

ST=7, SU = 10ST2 + TU2 = SU2

72 + TU2 = 102

49 + TU2 = 102

49 + TU2 = 100TU2 = 51

14.751TU

S

T U

A. Find x.

A.

B.

C.

D.

B. Find x.

A.

B.

C.

D.

Pythagorean Triple Pythagorean Triples are special sets of

numbers that all the numbers are positive integers.

A Pythagorean triple is 3, 4, and 5.

32 + 42 = 52

9 + 16 = 25

Pythagorean Triples under 100

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

( 9, 40, 41)(11, 60, 61)(12, 35, 37)

(13, 84, 85)(16, 63, 65)(20, 21, 29)

(28, 45, 53)(33, 56, 65)(36, 77, 85)

(39, 80, 89)(48, 55, 73)(65, 72, 97)

Formula:

Suppose that m and n are two positive integers, with m < n. Then n2 - m2, 2mn, and n2 + m2 is a Pythagorean triple.

p. 548

Find the length of the missing side.

8

610

Triple = 6, 8, 10

25 7

24

Triple = 7, 24, 25

12

13

5

Triple = 5, 12, 13

Use a Pythagorean triple to find x. Explain your reasoning.

Notice that 24 and 26 are multiples of 2: 24 = 2 ● 12 and 26 = 2 ● 13. Since 5, 12, 13 is a Pythagorean triple, the missing leg length x is 2 ● 5 or 10.

Answer: x = 10

Check: 242 + 102 = 262 Pythagorean Theorem?

676 = 676 Simplify.

A. 10

B. 15

C. 18

D. 24

Use a Pythagorean triple to find x.

p. 550

If you have the lengths of three sides of a triangle, you can use the converse of the Pythagorean Theorem to prove it is a right triangle.

You can also use side lengths to classify an acute or obtuse triangle.

p. 550

A. Determine whether 9, 12, and 15 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.

Step 1 Determine whether the measures can form a triangle using the Triangle Inequality Theorem.

9 + 12 > 15 9 + 15 > 12 12 + 15 > 9

The side lengths 9, 12, and 15 can form a triangle.Step 2 Classify the triangle by comparing the

square of the longest side to the sum of the squares of the other two sides.

c2 = a2 + b2 Compare c2 and a2 + b2.

?

152 = 122 + 92 Substitution?

225 = 225 Simplify and compare.Answer: Since c2 = a2 + b2, the triangle is a right triangle.

B. Determine whether 10, 11, and 13 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.

Step 1 Determine whether the measures can form a triangle using the Triangle Inequality Theorem.

10 + 11 > 13 10 + 13 > 11 11 + 13 > 10

The side lengths 10, 11, and 13 can form a triangle.

Step 2 Classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides.

c2 = a2 + b2 Compare c2 and a2 + b2.

?

132 = 112 + 102 Substitution?

169 < 221 Simplify and compare.

Answer: Since c2 < a2 + b2, the triangle is acute.

A. yes, acute

B. yes, obtuse

C. yes, right

D. not a triangle

A. Determine whether the set of numbers 7, 8, and 14 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.

• What is the Pythagorean Theorem?

a2 + b2 = c2

• Why is it important?

It is used to calculate the length of any side of a right triangle when the lengths of the other two sides are known.

• What is a Pythagorean Triple?

Pythagorean Triples are special sets of numbers that all the numbers are positive integers.

8-2 Assignment

Page 552, 8-28 even