Problem preview

G-SRT.1.a Warmup M2-045-A10-V01

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

Distinguish dilation effects on infinite lines and finite segments

Problem

Classify the exact dilation effect on an infinite line through center \(O\) under a dilation with \(k>0,k\ne~1\).

Unworked illustrative setup for an infinite line through center O(0,0), with k=2 and source points P(−2,−1) and Q(2,1); image points and the line-versus-length classification are not shown.

Big Picture

What this problem is really about

Set invariance does not mean every point stays fixed, and an infinite line is not a finite segment with a total length. We’ll parameterize the center-passing line, see how dilation rescales its parameter, show the image parameters still cover the whole line, and separate that result from the scaling of chosen finite subsegments.

Answer choices

Choose your path.

Preview only

Every point maps to the center, so the line collapses.

It becomes a longer or shorter finite segment.

The same finite segment is unchanged.

The infinite line is unchanged as a set because it passes through the center.

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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: Distinguish dilation effects on infinite lines and finite segments