Problem preview

G-SRT.1.a Warmup M2-045-A01-V01

Verify that dilations send non-center-passing lines to parallel lines and leave center-passing lines unchanged.

Find a line image under dilation, including the invariant case

Problem

Find the exact image line under this dilation: center \(O=(0,0)\),\(k=2\),line through \(P=(1,1)\),\(Q=(3,1)\).

Unworked dilation setup with center O(0,0), k=2, and source points P(1,1) and Q(3,1) on y=1; P′, Q′, and the image-line equation are not shown.

Big Picture

What this problem is really about

Two mapped points determine the exact image line, while center incidence determines whether that line is invariant or merely parallel. We’ll apply the dilation to both coordinates of two source points, derive their image equation, test whether the source contains the center, and compare direction and distinctness.

Answer choices

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Preview only

P′ = (2, 2) and Q′ = (6, 2), so the image line is y = 2.

P′ = (2, 1) and Q′ = (6, 1), so the image line is y = 1.

P′ = (2, 2) and Q′ = (6, 2), but no image equation can be determined.

Because the center is on y = 1, the image line is y = 1.

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Curriculum context

Course: Math II
Standard: G-SRT.1.a
Category: Geometry
Domain: Similarity, Right Triangles, and Trigonometry
Objective: Verify that dilations send non-center-passing lines to parallel lines and leave center-passing lines unchanged.
Problem type: Find a line image under dilation, including the invariant case