8.1 The Pythagorean Theorem and Its Converse We will learn to use the Pythagorean Theorem and its...

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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