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All courses Math II · G-SRT.1.b 46 of 73
Verify that dilations scale line segments by the dilation scale factor.

Find the image length of a segment after a dilation

Problem
Segment AB has length 8 units and is dilated by scale factor 3. What is the image length A'B'?
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Hint

Use image length = original length x scale factor with original length 8 and scale factor 3.

A dilation multiplies every length by \(k\), so \(A'B' = 3(AB)\).

Solution walkthrough

01

State the length rule

\[\text{image}~\text{length}=|k|(\text{original}~\text{length})\]

Dilation multiplies every linear measure by the absolute value of its scale factor. Here k=3 is positive, so |k|=3.

02

Source the values

\[AB=8~\text{units};~k=3\]

Eight is the preimage segment length, and 3 is the multiplicative dilation factor.

03

Substitute and compute

\[A'B'=3(8~\text{units})=24~\text{units}\]

The scale factor multiplies the full original length; it is not added to it.

04

Interpret and check

\[A'B'/AB=24/8=3=k\]

The image is three times as long, and the image-to-original ratio recovers the stated scale factor.

+

Another way

  1. Add three copies of the 8-unit original: 8+8+8=24.

!

Common mistake

Do not add 3 to 8. A scale factor is a ratio, so it multiplies the original length.