Question

a dilation maps △abc onto △abc. if ac = 24 cm, ac = 6 cm and bc = 30 cm, then bc = cm.

Explanation:

Step1: Find the scale factor

The scale factor $k$ of a dilation is given by the ratio of corresponding side - lengths. For sides $AC$ and $A'C'$, we have $k=\frac{A'C'}{AC}$. Given $AC = 24$ cm and $A'C'=6$ cm, then $k=\frac{6}{24}=\frac{1}{4}$.

Step2: Find the length of $B'C'$

Since the dilation is a similarity transformation, the ratio of corresponding side - lengths is the same for all pairs of corresponding sides. Let $B'C'$ be the side corresponding to $BC$. We know that $\frac{B'C'}{BC}=k$. Given $BC = 30$ cm and $k=\frac{1}{4}$, then $B'C'=k\times BC$. Substituting the values, we get $B'C'=\frac{1}{4}\times30 = 7.5$ cm.

Answer:

$7.5$