Question

a diagram is shown. what is the length, in centimeters, of $overline{mn}$? a 25 b 32 c 4.8 d 7.8

Explanation:

Step1: Check for similar - triangles

We can observe that \(\triangle ABC\) and \(\triangle MNC\) are similar. The ratios of corresponding sides of similar triangles are equal.

Step2: Set up the proportion

The ratio of the sides of \(\triangle ABC\) to \(\triangle MNC\) is \(\frac{AC}{NC}=\frac{BC}{MC}\). Here, \(AC = 15\) cm, \(NC = 6\) cm, \(BC = 10\) cm, \(MC = 4\) cm. Also, \(\frac{AB}{MN}=\frac{AC}{NC}\).

Step3: Calculate \(MN\)

We know that \(\frac{AB}{MN}=\frac{AC}{NC}\), substituting \(AB = 12\) cm, \(AC = 15\) cm, \(NC = 6\) cm. We get \(\frac{12}{MN}=\frac{15}{6}\). Cross - multiply: \(15\times MN=12\times6\). Then \(MN=\frac{12\times6}{15}=\frac{72}{15} = 4.8\) cm.

Answer:

C. 4.8