Q. 2STP - Algebra 2

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

Step-by-Step Solution

Answer

The length of $\overline{MQ}$ is 21.

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

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