Question

a triangle abc has vertices at a (6, 2), b (4, -2), and c (2, 4). the triangle is dilated about the origin with a scale factor of 1/2. which ordered pair will be the coordinates of one of the new vertices of triangle abc? answer choices: a) (3, 1) b) (1, 2) c) (6, 2) d) (4, -2) e) (2, -1) f) (1, 4) square dilated by scale factor 1/4 about the origin which statement shows a square dilated by a scale factor of 1/4 using the origin as the center of dilation? a) the square moves away from the origin and becomes 4 times larger. b) the square remains the same size but shifts to a new location. c) the square becomes 1/4 the size and moves closer to the origin. d) the square flips across the x - axis but keeps the same size.

Explanation:

Step1: Recall dilation formula

When a point $(x,y)$ is dilated about the origin with a scale - factor $k$, the new coordinates $(x',y')$ are given by $(x',y')=(k\times x,k\times y)$.

Step2: Calculate new vertices of triangle ABC

For vertex A(6,2) with $k = \frac{1}{2}$, $x'=\frac{1}{2}\times6 = 3$ and $y'=\frac{1}{2}\times2 = 1$, so the new vertex is (3,1).
For vertex B(4, - 2) with $k=\frac{1}{2}$, $x'=\frac{1}{2}\times4 = 2$ and $y'=\frac{1}{2}\times(-2)=-1$, so the new vertex is (2, - 1).
For vertex C(2,4) with $k=\frac{1}{2}$, $x'=\frac{1}{2}\times2 = 1$ and $y'=\frac{1}{2}\times4 = 2$, so the new vertex is (1,2).

Step3: Recall properties of dilation

When a figure is dilated with a scale - factor $k=\frac{1}{4}$ about the origin, each side - length of the figure is multiplied by $\frac{1}{4}$, so the figure becomes $\frac{1}{4}$ the size of the original figure and moves closer to the origin.

Answer:

For the first question: A) (3,1), E) (2, - 1), B) (1,2)
For the second question: C) The square becomes 1/4 the size and moves closer to the origin.