8.1 The Pythagorean Theorem and Its Converse

We will learn to use the Pythagorean Theorem and its converse.

The Pythagorean Theorem

• If a triangle is a right triangle, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

B

A C

ca

b

If ∆ABC is a right triangle

Then…leg² + leg² = hypotenuse²

a² + b² = c²

Vocabulary

• Pythagorean Triple: is a set of nonzero whole numbers a, b, and c that make the equation a² + b² = c² true.– Common Pythagorean Triples:

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

– If you multiply each number in a Pythagorean Triple by the same whole number, the three numbers that result also form a Pythagorean Triple.

3, 4, 5 6, 8, 10 3, 4, 5 9, 12, 15

Finding the Lengths of the Hypotenuse

Example: What is the length of the hypotenuse of ∆ABC? Do the side lengths of ∆ABC form a Pythagorean Triple? Explain?

A

B

C20

21

leg² + leg² = hypotenuse²

20² + 21² = c²400 + 441 = c² 841 = c² √(841) = c 29 = c

The side lengths 20, 21, and 29 form a Pythagorean Triple because they are whole numbers that satisfy a² + b² = c²

Finding the Length of the Hypotenuse

You Try: The legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse?

Do these side lengths form a Pythagorean Triple?

10² + 24² = c²100 + 576 = c² 676 = c² √(676) = c 26 = c

Yes, side lengths 10, 24, and 26 are whole numbers that satisfy a² + b² = c²

Finding the Length of a Leg

Example: What is the value of x? Express your answer in simplest radical form.

20

x

8

a² + b² = c² x² + 8² = 20² x² + 64 = 400 x² = 336 x = √(336) x = √(16·21) x = 4√(21)

Finding the Length of a Leg

You Try: The hypotenuse of a right triangle has length 12. One leg has length 6. What is the length of the other leg? Express your answer in simplest radical form.a² + b² = c² x² + 6² = 12² x² + 36 = 144 x² = 108 x = √(108) x = √(36·3) x = 6√(3)

Finding Distance

Example: Dog agility courses often contain a seesaw obstacle, as shown below. To the nearest inch, how far above ground are the dog’s paws when the seesaw is parallel to the ground?

36 in.

26 in.

a² + b² = c² x² + 26² = 36² x² + 676 = 1296 x² = 620 x = √(620) x = √(4·155) x = 2√(155) or ≈ 24.8997992

The dog’s paws are 25 inches from the ground when the seesaw is parallel to the ground.

Finding Distance

You Try: The size of a computer monitor is the length of its diagonal. You want to buy a 19-inch monitor that has a height of 11 inches. What is the width of the monitor? Round to the nearest tenth of an inch.

1911

a² + b² = c² x² + 11² = 19² x² + 121 = 361 x² = 240 x = √(240) x = √(16·15) x = 4√(15) or ≈ 15.5

The monitor is 15.5 inches wide.

The Converse of the Pythagorean Theorem

• If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. B

A C

ca

b

If a² + b² = c²

Then…∆ABC is a right triangle.

Identifying a Right Triangle

Example: A triangle has side lengths 85, 84, and 13. Is the triangle a right triangle? Explain.

a² + b² = c² 13² + 84² = 85²169 + 7056 = 7225 7225 = 7225

Yes, the triangle is a right triangle because 13² + 84² = 85².

** The longest side of the triangle always needs to be plugged in for c, the hypotenuse.

Identifying a Right Triangle

You Try: A right triangle has side lengths 16, 48, and 50. Is the triangle a right triangle? Explain.

a² + b² = c² 16² + 48² = 50²256 + 2304 = 2500 2560 ≠ 2500

No, the triangle is not a right triangle because 16² + 48² ≠ 50².

Pythagorean Inequalities Theorem

• 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, then the triangle is obtuse.

A

B

C

c

b

a

If…c² > a² + b²

Then…∆ABC is obtuse

Pythagorean Inequalities Theorem

• If the square 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, then the triangle is acute.

A

B

C

c

b

a

If…c² < a² + b²

Then…∆ABC is acute.

Classifying a Triangle

Example: A triangle has side lengths 6, 11, and 14. Is it acute, obtuse, or right?

c² > a² + b² 14² > 6² + 11² 196 > 36 + 121196 > 157

This triangle is obtuse since 14² > 6² + 11².

Classifying a Triangle

You Try: Is a triangle with side lengths 7, 8, and 9 acute, obtuse, or right?

c² < a² + b² 9² < 7² + 8² 81 < 49 + 6481 < 113

This triangle is acute since 9² < 7² + 8².

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