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All courses Geometry · G-SRT.3 33 of 57
Use similarity transformations to establish the AA similarity criterion.

Use AA similarity to find a missing side

Problem
The triangles are similar by \(AA\). Use this information to find the missing side: corresponding sides \(4\) and \(12\) match, and \(5\) corresponds to \(x\).
Neutral AA similar-triangle side reference: 4 corresponds to 12 and 5 corresponds to x, without completed triangles. Open full size
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Hint

Use the matching sides \(4\) and \(12\) to find the scale factor first.

Similar triangles have proportional corresponding sides, so once you know one scale factor, you use it on every matching side pair.

Solution walkthrough

01

State why side proportionality is available

\[\text{two}~\angle~\text{pairs}~\text{are}~\text{congruent}~->~\text{triangles}~\text{similar}~by~AA\]

AA establishes similarity, so corresponding side lengths must share one scale factor.

02

Pair the supplied sides

\[4<->12;~5<->x\]

The diagram and correspondence match the side of length 4 with 12 and the side of length 5 with x.

03

Find and apply the scale factor

\[k=12/4=3;~x=3(5)=15\]

The second triangle is three times the first, so its side corresponding to 5 has length 15.

04

Verify with equal ratios

\[12/4=15/5=3\]

Both image-to-preimage ratios agree, confirming the missing side while respecting the AA correspondence.

+

Another way

  1. Solve 4/12=5/x; cross-multiplication gives 4x=60, so x=15.

!

Common mistake

AA proves the triangles are similar, but it does not make corresponding sides equal. Their lengths differ by the common scale factor.

Answer AA proportion: corresponding sides 4 and 12 give scale factor 12/4=3 from the first triangle to the second. Side 5 corresponds to x, so x=3·5=15. The missing side is 15.
AA correspondence fixes the side proportion and missing length.

Solution walkthrough video