Question

2
select the true statement.
a. if line cd contains the points c(2,2) and d(4,4) and is dilated with the center of dilation at the origin, then the dilation would be perpendicular to line cd.
b. if line gh contains the points g(-2,-2) and h(-4,-4) and is dilated with the center of dilation at the origin, then the dilation would be perpendicular to line gh.
c. if line ab contains the points a(0,2) and b(0,4) and is dilated with the center of dilation at the origin, then the dilation is on the x - axis.
d. if line ef contains the points e(2,0) and f(4,0) and is dilated with the center of dilation at the origin, then the dilation is on the x - axis.

Explanation:

Step1: Analyze Option A

Line CD has points \( C(2,2) \) and \( D(4,4) \). The slope of CD is \( \frac{4 - 2}{4 - 2}=1 \), so it's a line with slope 1 (y = x). Dilation from the origin scales points, but the dilated line will be parallel to CD (same slope), not perpendicular. So A is false.

Step2: Analyze Option B

Line GH has points \( G(-2,-2) \) and \( H(-4,-4) \). Slope is \( \frac{-4 - (-2)}{-4 - (-2)} = 1 \), equation \( y = x \). Dilation from origin keeps the line parallel (same slope), not perpendicular. So B is false.

Step3: Analyze Option C

Line AB has points \( A(0,2) \) and \( B(0,4) \), which is a vertical line (x = 0, y - axis). Dilation from origin: points become \( (0, 2k) \) and \( (0, 4k) \), still on y - axis, not x - axis. So C is false.

Step4: Analyze Option D

Line EF has points \( E(2,0) \) and \( F(4,0) \), which is a horizontal line on x - axis (y = 0). Dilation from origin scales x - coordinates: \( (2k, 0) \) and \( (4k, 0) \), still on x - axis. So D is true.

Answer:

D. If line EF contains the points E(2,0) and F(4,0) and is dilated with the center of dilation at the origin, then the dilation is on the x - axis.