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All courses Math II · G-GMD.5 39 of 73
Apply scale factors to length, area, and volume using k, k^2, and k^3 relationships.

Find the new length after a dilation by scale factor k

Problem
An object with original length 8 is dilated by scale factor 3. Find the new length.
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Hint

Multiply the original length by the scale factor.

Under a dilation, \(new length = original length \times scale factor\).

Solution walkthrough

01

State the linear dilation rule

\[\text{new}~\text{length}=k~\text{times}~\text{original}~\text{length}\]

A dilation multiplies every linear measure by its scale factor k. Length uses the first power of k.

02

Map the given values

\[\text{original}~\text{length}=8;~k=3\]

The starting length is 8 and the dilation scale factor is 3, so the image must be three copies of the original length.

03

Calculate the image length

\[\text{new}~\text{length}=3(8)=24\]

Multiplying 8 by 3 gives 24.

04

Interpret and check

\[24/8=3\]

The ratio of new length to original length is 3, matching the stated scale factor.

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Another way

  1. View scale factor 3 as three equal copies: 8+8+8=24.

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Common mistake

Do not add 3 to 8. A scale factor is a multiplicative ratio, so the new length is 3 times 8.