Warmup:Solve for y

8.2 Special Right Triangles LEQ: What are special right triangles and how do we use them?

The “root” of it all… Calculate the value of each hypotenuse in a triangle as a reduced radical… Do you notice a trend? Is there an easier way to calculate (shortcuts)? (hint: label sides with numbers)

Side lengths of Special Right Triangles Right triangles whose angle measures are 45°-45°-90° or 30°- 60°-90° are called special right triangles. The theorems that describe these relationships of side lengths of each of these special right triangles follow….

Theorem 8-5: 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 = leg ∙ √2

Finding the hypotenuse in a 45°- 45°-90° Triangle Ex 1: Find the value of x at the right. Ex 2: Find the length of the hypotenuse of a triangle with legs of length 5√3. 2√16 x 45 °

Finding a leg in a 45°-45°-90° Triangle Ex. 3 Find the value of x and y. 5 x y 45 °

Theorem 8-6: 30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle: –the hypotenuse is twice as long as the shorter leg –the longer leg is √3 times as long as the shorter leg. x√3 60 ° 30 ° Hypotenuse = 2 ∙ (shorter leg) Longer leg = (shorter leg) ∙ √3

Alternatives? OF COURSE!! If you don’t care for shortcuts, you may memorize… 1 1 √2 1 2 √

Finding side lengths in a 30°-60°- 90° Triangle Ex. 1 Find the values of s and t and the right. Ex. 2 Find the value of each variable. 30 ° 60 ° 30 ° 60 ° 8 x y

Real World Connection The yield sign is an equilateral triangle. It has a height of 18√3. Find x and y. 18√3 y inch. x inch.