Question

what is the result of dilating a figure with a scale factor of 1?
a. the figure is enlarged by twice
b. the figure is shifted to a new position
c. the figure remains the same size
d. the figure is reduced by half

what is the role of the center of dilation in a transformation?
a. it ensures angles remain congruent
b. it determines the orientation of the image
c. it acts as the fixed point from which all distances are scaled
d. it acts as the fixed point from which all angles are enlarged

dilate △def with vertices d(2, 1), e(4, 5), f(6, 3) by a scale factor k = 2 with the origin as the center of dilation. what are the coordinates of f?
a. (12, 3)
b. (6, 12)
c. (12, 6)
d. (3, 12)

Explanation:

First Question:

Dilation with a scale factor of 1 means each coordinate (or length) is multiplied by 1, so the figure's size doesn't change. Option a is for scale factor 2, option b is translation (not dilation), option d is for scale factor 1/2.

The center of dilation is the fixed point from which distances are scaled (multiplied by the scale factor). Angles remain congruent due to similarity, not the center (a wrong). Orientation isn't determined by the center (b wrong). Angles aren't enlarged from the center (d wrong).

Step1: Recall dilation rule (origin as center)

When dilating with center at origin and scale factor \( k \), the new coordinates \((x', y')\) of a point \((x, y)\) are \( x' = kx \), \( y' = ky \).

Step2: Apply to point \( F(6, 3) \) with \( k = 2 \)

For \( x \)-coordinate: \( x' = 2 \times 6 = 12 \)
For \( y \)-coordinate: \( y' = 2 \times 3 = 6 \)
So \( F' \) is \( (12, 6) \).

Answer:

c. The figure remains the same size

Second Question: