Question

Each triangle is a 30-60-90 triangle, and the hypotenuse of one triangle is the longer leg of an adjacent triangle. the hypotenuse of the largest triangle is 8 centimeters. what is the number of centimeters in the length of the longer leg of the smallest triangle? express your answer as a common fraction.

Answer 1

Let the first right triangle be ABC where A is the right angle and BC the hypotenuse of length a.
If BC is the longer leg of the second right triangle and CD is the shorter leg, the longer hypotenuse is BD of length x, and the right angle is BCD. BD=x=8cm. CBD=BCA=30 and BD is parallel to AC, making BDCA a trapezoid. Also a/x=cos30=√3/2.
So a=8×√3/2=4√3 cm. the longer leg of ABC, the smaller triangle, is AC. AC/a=cos30=√3/2, so AC=a√3/2=(4√3)√3/2=6 cm.