Question

triangle rst has the coordinates r (-2, 2), s (2, 7), and t (6, 2). which of the following sets of points represents a dilation from the origin of triangle rst?
a. r (4, 8), s (8, 13), t (12, 8)
b. r (-12, 12), s (12, 42), t (36, 12)
c. r (-12, 2), s (2, 42), t (36, 2)
d. r (-12, 2), s (12, 7), t (36, 2)

Explanation:

Step1: Define dilation rule

A dilation from the origin multiplies each coordinate of a point $(x,y)$ by a scale factor $k$, so the transformed point is $(kx, ky)$.

Step2: Test Option A

Check if all points follow the rule:
For $R(-2,2)$ to $R'(4,8)$: $k=\frac{4}{-2}=-2$, but $\frac{8}{2}=4$. No consistent $k$.

Step3: Test Option B

Check if all points follow the rule:
For $R(-2,2)$ to $R'(-12,12)$: $k=\frac{-12}{-2}=6$, $\frac{12}{2}=6$.
For $S(2,7)$ to $S'(12,42)$: $k=\frac{12}{2}=6$, $\frac{42}{7}=6$.
For $T(6,2)$ to $T'(36,12)$: $k=\frac{36}{6}=6$, $\frac{12}{2}=6$.
Consistent scale factor $k=6$.

Step4: Verify other options (optional)

Option C: $R(-2,2)$ to $R'(-12,2)$: $x$-coordinate scaled by 6, $y$-coordinate unchanged. Not dilation.
Option D: $S(2,7)$ to $S'(12,7)$: $x$-coordinate scaled by 6, $y$-coordinate unchanged. Not dilation.

Answer:

B. R' (-12, 12), S' (12, 42), T' (36, 12)