Question

the height of trapezoid vwxz is ( 8sqrt{3} ) units. the upper base, ( overline{vw} ), measures 10 units. use the ( 30^{circ}-60^{circ}-90^{circ} ) triangle theorem to find the length of ( overline{yx} ). once you you know the length of ( overline{yx} ), find the length of the lower base, ( overline{zx} ).
14 units
( 10 + 4sqrt{3} ) units
18 units
( 10 + 8sqrt{3} ) units

Explanation:

Step1: Find the length of \( YX \)

In a \( 30^{\circ}-60^{\circ}-90^{\circ} \) triangle \( WYX \), the side opposite the \( 30^{\circ} \) angle is \( WY = 8\sqrt{3} \).
Let the side opposite \( 30^{\circ} \) be \( a \), the side opposite \( 60^{\circ} \) be \( a\sqrt{3} \), and the side opposite \( 90^{\circ} \) be \( 2a \).
Here \( a = 8 \) (since \( WY=a\sqrt{3}=8\sqrt{3}\Rightarrow a = 8 \)), so \( YX=8 \) (the side opposite \( 30^{\circ} \) in the right - triangle formed by the height \( WY \)).

Step2: Calculate the length of \( ZX \)

We know that \( ZX=ZY + YX \). Since \( ZY = VW = 10 \) (in a trapezoid, the segment \( ZY \) is equal to the upper base \( VW \) as they are between two parallel lines and the two right - angles).
So \( ZX=10 + 8=18 \) units.

Answer:

C. 18 units