Apply the Pythagorean Theorem

Chapter 7.1

Sides of a Right Triangle

• Hypotenuse – the side of a right triangle opposite the right angle and the longest side.

• Legs – the sides of a right triangle that are not the hypotenuse.

The Pythagorean Theorem

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

Proving the Pythagorean Theorem

• There are multiple ways of proving the Pythagorean Theorem.

• If you can find a legitimate proof of the Pythagorean theorem you may earn 2 bonus points.

Using the Pythagorean Theorem

222 cba 6 8 x²

36 + 64 = x²

100= x²2100 x x10

Find the missing leg

222 cba x 3 5²

x²+ 9 = 25

x² = 16162 x 4x

Find the Area of the TriangleWhat is the formula for the area of a triangle?

A = ½bh

How will we find the height?222 135 h

16925 2 h1442 h 12h

bhA2

1 )12)(10(

2

1 60

Find the Area of the Triangle

222 124 h14416 2 h

1282 h

28h

bhA2

1 )28)(8(

2

1 232

Pythagorean Triples

• A Pythagorean Triple is a set of 3 positive integers or whole numbers that satisfies the Pythagorean theorem.

Is it a Pythagorean Triple?

• 3, 4, and 5

• 21, 28, and 35

• 30, 72, and 91

• 14, 48, and 50

2525

25169

543 222

yes

yes

no

yes

If I am given 2 sides of a right triangle make up the sides of Pythagorean triple,

how do you find the missing side?

There are 2 possible scenarios:1. You are given both legs of the right triangle

and need to solve for the hypotenuse.

2. You are given one leg and one hypotenuse and need to solve for the other leg.

Given 2 sides: 20 and 25• Scenario 1. • 20 and 25 are the legs• 202 + 252 = x2

• 400 + 625 = x2

• 1025 = x2

• 32.02 = x• This cannot be the

answer for a Pythagorean triple because it is not a whole number.

• Scenario 2. • 20 and 25 are the leg

and hypotenuse• 202 + x2 = 252

• 400 + x2 = 625• x2 = 225• X = 25• This can be the answer

for a Pythagorean triple because it is not a whole number.

Given 2 sides: 28 and 96

Find the area when given a leg and the hypotenuse

1. Find the other leg by plugging the known values into the Pythagorean Theorem.

2. Use the 2 legs in the formula for area of a triangle A = ½BH

Find the area when given a leg and a hypotenuse.

• L=8 and h= 16• Aò+8ò=16ò• Aò+64=256• Aò=192

1922 a

42.5519282

1

2

1bh

Find the area when given a leg and a hypotenuse.

• L=13 and h= 17• Aò+13ò=17ò• Aò+169=289• Aò=120

1202 a

2.71120132

1

2

1bh

• #3 – 7, 11- 16, 18 - 23