2. Goal 2: To use the properties of 30°-60°-90° triangles Warm-up exercises Solve the equation for the missing variable. Assume all variables are positive. Express the answer in simplified radical form. 1. c 2 = 6 2 + 6 2 2. c 2 – 4 2 = 4 2 3. a 2 + 8 2 = 256 4. + b 2 = 1296

3. Solutions 1. 2. 3. 4. 18

4. Start with an equilateral, equiangular triangle. 60 30 60 30 60 30 Draw an angle bisector Isolate one 30-60-90 triangle formed. xx 2x2x2x2x 2x2x x To find the third, unknown side of the right triangle, use the Pythagorean Theorem. The 30-60-90 Triangle

5. 30 60 x 2x2x b a 2 + b 2 = c 2 Pythagorean Theorem x 2 + b 2 = (2x) 2 Substitution Property of Equality x 2 + b 2 = 4x 2 Simplify b 2 = 3x 2 Subtraction Property of Equality b = Simplify

6. Theorem: In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg and the longer leg is times as long as the shorter leg. 30 60 x 2x2x

7. Short Leg Hypotenuse Long Leg Hypotenuse Long Leg Divide by 2 Divide by Multiply by 2 Multiply by 30-60-90 Triangles Always solve for the short leg first.

8. Ex. Find the values of the variable(s). 1. 2. 30 60 6 s 5 t s 30 60

9. Ex. A tipping platform is a ramp used to unload trucks, as shown in the figure. How high is the end of an 80 foot ramp when it is tipped by a 30 degree angle? by a 45 degree angle?

10. Ex. The road sign is shaped like an equilateral triangle. Estimate the area of the sign by finding the area of the equilateral triangle.

11. Assignment Assignment: pp. 369-71 #12-30, 33-39