Question

triangles abc and def are similar triangles. what is the length of side n? *not drawn to scale a. 6 cm b. 6 1/4 cm c. 7 1/4 cm d. 8 cm

Explanation:

Step1: Find the scale - factor

For similar triangles, the ratio of corresponding sides is the same. Let's find the ratio of the sides of $\triangle ABC$ and $\triangle DEF$ using the known corresponding sides. Consider the sides $AB = 9\frac{3}{4}=\frac{9\times4 + 3}{4}=\frac{39}{4}$ cm and $DE=3\frac{1}{4}=\frac{3\times4 + 1}{4}=\frac{13}{4}$ cm. The scale - factor $k=\frac{AB}{DE}=\frac{\frac{39}{4}}{\frac{13}{4}} = 3$.

Step2: Calculate the length of side $n$

We know that side $BC = 18\frac{3}{4}=\frac{18\times4+3}{4}=\frac{75}{4}$ cm corresponds to side $EF=n$. Since the scale - factor is 3, we can set up the proportion $\frac{BC}{EF}=3$. So, $n=\frac{BC}{3}$. Substituting $BC = \frac{75}{4}$ cm, we get $n=\frac{\frac{75}{4}}{3}=\frac{75}{4}\times\frac{1}{3}=\frac{75}{12}=\frac{25}{4}=6\frac{1}{4}$ cm.

Answer:

B. $6\frac{1}{4}$ cm