Chapter 9 Summary

Similar Right Triangles If the altitude is drawn to the hypotenuse of a right triangle, then the 3 triangles are all similar.

A B C D

Find QS.

Solve for x

A B C D

Find XZ

Pythagorean Theorem In a right triangle,

Acute, Right, Obtuse Triangles Acute Right Obtuse

Pythagorean Triples Any 3 whole numbers that satisfy the pythagorean theorem. – Example: 3, 4, 5 – Nonexample: anything with a decimal or square root!

45 °-45°-90° Triangle Theorem In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg. x√2 45 ° Hypotenuse = √2 ∙ leg

30 °-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg. x√3 60 ° 30 ° Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg

Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle Find the value of x By the Triangle Sum Theorem, the measure of the third angle is 45 °. The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg. 33 x 45 °

Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle Hypotenuse = √2 ∙ leg x = √2 ∙ 3 x = 3√2 33 x 45 ° 45°-45°-90° Triangle Theorem Substitute values Simplify

Ex. 2: Finding a leg in a 45°-45°-90° Triangle Find the value of x. Because the triangle is an isosceles right triangle, its base angles are congruent. The triangle is a 45°-45°-90° right triangle, so the length of the hypotenuse is √2 times the length x of a leg. 5 x x

Ex. 2: Finding a leg in a 45°-45°-90° Triangle 5 x x Statement: Hypotenuse = √2 ∙ leg 5 = √2 ∙ x Reasons: 45°-45°-90° Triangle Theorem 5 √2 √2x √2 = 5 x= 5 x= 5√2 2 x= Substitute values Divide each side by √2 Simplify Multiply numerator and denominator by √2 Simplify

Ex. 3: Finding side lengths in a 30°-60°-90° Triangle Find the values of s and t. Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg. 30 ° 60 °

Ex. 3: Side lengths in a 30°-60°-90° Triangle Statement: Longer leg = √3 ∙ shorter leg 5 = √3 ∙ s Reasons: 30°-60°-90° Triangle Theorem 5 √3 √3s √3 = 5 s= 5 s= 5√3 3 s= Substitute values Divide each side by √3 Simplify Multiply numerator and denominator by √3 Simplify 30 ° 60 °

The length t of the hypotenuse is twice the length s of the shorter leg. Statement: Hypotenuse = 2 ∙ shorter leg Reasons: 30°-60°-90° Triangle Theorem t2 ∙ 5√3 3 = Substitute values Simplify 30 ° 60 ° t 10√3 3 =

Parts of the Triangle SohCahToa

opposite hypotenuse adjacent hypotenuse opposite adjacent

Find the values of the three trigonometric functions of 

Finding a missing side

Finding a missing angle We can find an unknown angle in a right triangle, as long as we know the lengths of two of its sides. – Use Trig Inverse – sin -1 – cos -1 – tan -1

What is sin -1 ? But what is the meaning of sin -1 … ? Well, the Sine function "sin" takes an angle and gives us the ratio “opposite/hypotenuse” But in this case we know the ratio “opposite/hypotenuse” but want to know the angle. So we want to go backwards. That is why we use sin -1, which means “inverse sine”.

Find the missing angle