Question

In the figure, $\Delta \mathbf{MOQ}$is similar to$\Delta \mathbf{NOP}.$What is the length of$\overline{\mathrm{MQ}}?$

(A) 12

(B) 12.5

(C) 19

(D) 21

Short answer

The length of$\overline{MQ}$ is 21.

Step-by-step solution

Step 1. Sketch the given diagram

Step 2. Evaluate ∠NOP

Let$\angle NOP=\theta .$

$\begin{array}{c}\sin \theta =\frac{Perpendicular}{Hypothesis}\\ =\frac{4.5}{9}\\ =\frac{1}{2}\\ \theta ={30}^{\circ }\end{array}$

Step 3. Evaluate the length of MQ¯

Since $\angle NOP={30}^{\circ },$$\angle MOQ={30}^{\circ }.$

$\begin{array}{c}\sin {30}^{\circ }=\frac{Perpendicular}{Hypothesis}\\ \frac{1}{2}=\frac{OQ}{MQ}\\ =\frac{OP+PQ}{MQ}\\ \frac{1}{2}=\frac{6+4.5}{MQ}\\ MQ=2\times 10.5\\ =21\end{array}$