Question

in the figure below, $\triangle abc \sim \triangle mnp$. $\overline{ad}$ is a median segment of $\triangle abc$ and $\overline{mq}$ is a median segment of $\triangle mnp$. suppose $ad = 20$, $bc = 16$, and $np = 8$. find $mq$. (note that the figure is not drawn to scale.)

Explanation:

Step1: Recall the property of similar triangles

For similar triangles, the ratio of corresponding medians is equal to the ratio of corresponding sides. So, if \(\triangle ABC \sim \triangle MNP\), then \(\frac{AD}{MQ}=\frac{BC}{NP}\).

Step2: Substitute the given values

We know that \(AD = 20\), \(BC = 16\), and \(NP = 8\). Substituting these values into the proportion \(\frac{AD}{MQ}=\frac{BC}{NP}\), we get \(\frac{20}{MQ}=\frac{16}{8}\).

Step3: Solve for \(MQ\)

First, simplify \(\frac{16}{8}=2\). So the equation becomes \(\frac{20}{MQ}=2\). Cross - multiplying gives \(16\times MQ=20\times8\)? Wait, no, from \(\frac{20}{MQ}=2\), we can multiply both sides by \(MQ\) to get \(20 = 2\times MQ\), then divide both sides by 2: \(MQ=\frac{20}{2}=10\).

Answer:

\(10\)