Geometry question

Question

Given isosceles trapezoid ABCD with BA / / CD . Determine the perimeter of isosceles trapezoid ABCD in exact form when BC=4 and BA=9. (BC is sort of a hypotenuse for the trapezoid by the way).

and angle C measures 30 degrees.

Answer 1

Below is a drawing of the problem.

Since the trapezoid is isosceles, since BC is 4 and angle C is 30, then AD is 4 and angle D is 30.

To find the perimeter we just need to find the length of the segment x, which is equal for both sides of the trapezoid because it is isosceles.

To find x we see that we can form a right triangle and x is a leg of a 30-60-90 right triangle that would be opposite the 60 degree angle.

Since ratio of sides of a 30-60-90 right triangles are 1: sqrt(3):2 and the hypotenuse is 4, that leg would be 2* sqrt (3).

Thus, the perimeter is 4 + 4 + 9 + 9 + 2*sqrt(3) + 2*sqrt(3)= 26 + 4*sqrt(3) = = 32.93.

9

B------------------------A

/ | | \

4 / | | \ 4

/ | | \

/ | | \

/ | | \

/ 30 | | 30 \

-------------------------------------

C x | 9 | x D