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 (-12, 2), s (12, 7), t (36, 2)
b. r (-12, 12), s (12, 42), t (36, 12)
c. r (-12, 2), s (2, 42), t (36, 2)
d. r (4, 8), s (8, 13), t (12, 8)

Explanation:

Step1: Recall the dilation formula

If a point \((x,y)\) is dilated from the origin by a scale factor \(k\), the new point is \((kx,ky)\).

Step2: Check each option for a consistent scale factor

• Option A:

For \(R(-2,2)\) to \(R'(-12,2)\), \(x\) - scale factor \(k_x=\frac{-12}{-2} = 6\), \(y\) - scale factor \(k_y=\frac{2}{2}=1\). Since \(k_x
eq k_y\), this is not a dilation.

• Option B:

For \(R(-2,2)\) to \(R'(-12,12)\), \(k=\frac{-12}{-2}=\frac{12}{2} = 6\); for \(S(2,7)\) to \(S'(12,42)\), \(k=\frac{12}{2}=\frac{42}{7}=6\); for \(T(6,2)\) to \(T'(36,12)\), \(k=\frac{36}{6}=\frac{12}{2}=6\).

• Option C:

For \(R(-2,2)\) to \(R'(-12,2)\), \(k_x=\frac{-12}{-2}=6\), \(k_y = 1\). Since \(k_x
eq k_y\), this is not a dilation.

• Option D:

For \(R(-2,2)\) to \(R'(4,8)\), \(k_x=\frac{4}{-2}=-2\), \(k_y=\frac{8}{2} = 4\). Since \(k_x
eq k_y\), this is not a dilation.

Answer:

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