Question

19 mark for review note: figure not drawn to scale. in the figure shown, $overline{wz}$ and $overline{xy}$ intersect at point $q$. $yq = 63$, $wq = 70$, $wx = 60$, and $xq = 120$. what is the length of $overline{yz}$?

Explanation:

Step1: Prove triangles are similar

Since $\angle WQX=\angle YQZ$ (vertical - angles) and $\angle W=\angle Y$ (given $\angle W = \angle Y=a^{\circ}$), $\triangle WQX\sim\triangle YQZ$ by the AA (angle - angle) similarity criterion.

Step2: Set up proportion

For similar triangles $\triangle WQX$ and $\triangle YQZ$, the ratios of corresponding sides are equal. That is, $\frac{WQ}{YQ}=\frac{WX}{YZ}=\frac{XQ}{ZQ}$. We want to find $YZ$. Using the proportion $\frac{WQ}{YQ}=\frac{WX}{YZ}$, substituting the given values $WQ = 70$, $YQ = 63$, and $WX = 60$. We get $\frac{70}{63}=\frac{60}{YZ}$.

Step3: Solve for $YZ$

Cross - multiply: $70\times YZ=63\times60$. Then $YZ=\frac{63\times60}{70}$. Calculate $63\times60 = 3780$ and $\frac{3780}{70}=54$.

Answer:

54