Question

Dilate the figure by a scale factor of $0.5$ with the origin as the center of dilation. What are the vertices of the image?

• A. $\text{F}\text{'}\left(3,2.5\right),\text{G}\text{'}\left(5,0.5\right),\text{H}\text{'}\left(2,1\right)$
• B. $\text{F}\text{'}\left(2.5,3\right),\text{G}\text{'}\left(0.5,5\right),\text{H}\text{'}\left(0.5,1\right)$
• C. $\text{F}\text{'}\left(3,2.5\right),\text{G}\text{'}\left(5,0.5\right),\text{H}\text{'}\left(1,0.5\right)$
• D. $\text{F}\text{'}\left(-3,-2.5\right),\text{G}\text{'}\left(-5,-0.5\right),\text{H}\text{'}\left(-1,-0.5\right)$

Answer 1

The correct option is C

$\text{F}\text{'}\left(3,2.5\right),\text{G}\text{'}\left(5,0.5\right),\text{H}\text{'}\left(1,0.5\right)$

Find the vertices of the image:

Explanation for correct option: Option (C) :

Consider the given figure.

The formula for dilations are $\left(x\times t,y\times t\right)$, where $t$ is the scale factor.

In the given figure, the coordinate of the vertex $\text{F}$ is $\left(6,5\right)$.

Using the formula for dilations and the new vertex $\text{F}\text{'}$ can be expressed below :

$\left(6\times 0.5,5\times 0.5\right)=\left(3,2.5\right)$

So, the coordinate of the vertex $\text{F'}$ is $\left(3,2.5\right)$.

In the given figure, the coordinate of the vertex $\text{G}$ is $\left(10,1\right)$.

Using the formula for dilations and the new vertex $\text{G}\text{'}$ can be expressed below :

$\left(10\times 0.5,1\times 0.5\right)=\left(5,0.5\right)$

So, the coordinate of the vertex $\text{G'}$ is $\left(5,0.5\right)$.

In the given figure, the coordinate of the vertex $\text{H}$ is $\left(2,1\right)$.

Using the formula for dilations and the new vertex $\text{H}\text{'}$ can be expressed below :

$\left(2\times 0.5,1\times 0.5\right)=\left(1,0.5\right)$

So, the coordinate of the vertex $\text{H'}$ is $\left(1,0.5\right)$.

Therefore, the new vertices are $\text{F}\text{'}\left(3,2.5\right),\text{G}\text{'}\left(5,0.5\right),\text{H}\text{'}\left(1,0.5\right)$.

Therefore, option (C) is correct.

Explanation for incorrect options:

Since, the vertices obtained above is not equivalent to the value mentioned in option (A), option (B), and option (D).

Hence, option (A), option (B), and option (D) are incorrect options.

Hence, option (C) is the correct answer.