Triangles

9.2 The Pythagorean Theorem

In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b are legs and c is the hypotenuse. a c b

Pythagorean Triples Pythagorean Triple  When the sides of a right triangle are all integers it is called a Pythagorean triple.  3,4,5 make up a Pythagorean triple since = 5 2.

Example 1 Find the unknown side lengths. Determine if the sides form a Pythagorean triple y x

Example 2 Find the unknown side lengths. Determine if the sides form a Pythagorean triple. 100 q p

Example 3 Find the unknown side lengths. Determine if the sides form a Pythagorean triple. 2 3 e d

Example 4 Find the unknown side lengths. Determine if the sides form a Pythagorean triple. 5 g 8 f

9.3 The Converse of the Pythagorean Theorem

a c 2 = a 2 + b 2 b a b If a and b stay the same length and we make the angle between them smaller, what happens to c?

If a and b stay the same length and we make the angle between them bigger, what happens to c? a c 2 = a 2 + b 2 b a b

Classifying Triangles Let c be the biggest side of a triangle, and a and b be the other two side. If c 2 = a 2 + b 2, then the triangle is right. If c 2 < a 2 + b 2, then the triangle is acute. If c 2 > a 2 + b 2, then the triangle is obtuse. *** If a + b is not greater than c, a triangle cannot be formed.

Example 1 Determine what type of triangle, if any, can be made from the given side lengths. 7, 8, 12 11, 5, 9

Example 2 Determine what type of triangle, if any, can be made from the given side lengths. 5, 5, 5 1, 2, 3

Example 3 Determine what type of triangle, if any, can be made from the given side lengths. 16, 34, 30 9, 12, 15

Example 4 Determine what type of triangle, if any, can be made from the given side lengths. 13, 5, 7 13, 18, 22

Example 5 Determine what type of triangle, if any, can be made from the given side lengths. 4, 8, 5,, 5

9.4 Special Right Triangles

45º-45º-90º Triangles Solve for each missing side. What pattern, if any do you notice?

45º-45º-90º Triangles

300 ½ ½

45º-45º-90º Triangles x x

In a 45º-45º-90º triangle, the hypotenuse is times each leg. x x

30º-60º-90º Triangles Solve for each missing length. What pattern, if any do you notice? 10

30º-60º-90º Triangles 8 8 8

6 6 6

50

30º-60º-90º Triangles 2x

30º-60º-90º Triangles In a 30º-60º-90º triangle, the hypotenuse is twice as long as the shortest leg, and the longer leg is times as long as the shorter leg. 2x x 30º 60º

Example 1 Find each missing side length. 45º 15 45º 6

Example 2 45º 12 30º 18

Example 3 30º 12 30º 44

2x x 30º 60º x x

9.5 Trigonometric Ratios

Warm Up Name the side opposite angle A. Name the side adjacent to angle A. Name the hypotenuse. A C B

Trigonometric Ratios The 3 basic trig functions and their abbreviations are  sine = sin  cosine = cos  tangent = tan

SOH CAH TOA sin = opposite side hypotenuse cos = adjacent side hypotenuse tan = opposite side adjacent side SOH CAH TOA

Example 1 Find each trigonometric ratio.  sin A  cos A  tan A  sin B  cos B  tan B 3 4 C A 5 B

Example 2 Find the sine, the cosine, and the tangent of the acute angles of the triangle. Express each value as a decimal rounded to four decimal places E D 25 F

9.6 Solving Right Triangles

Example 1 Find the value of each variable. Round decimals to the nearest tenth. 25º 8 a

Example 2 42º 40 b

Example 3 20º c 8

Example 4 17º 10 c