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All courses Math II · G-SRT.2 47 of 73
Use similarity transformations to decide similarity and explain angle equality and side proportionality in triangles.

Decide SSS similarity from three explicit ratios

Problem
Decide triangle similarity from T1 sides3,4,5; T2 sides6,8,10.
Neutral triangle side-length reference listing T1 sides 3, 4, 5 and T2 sides 6, 8, 10, without drawn triangles or a similarity cue. Open full size
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Hint

Verify each triple satisfies the strict triangle inequality.

Sort both side triples and compare shortest-to-shortest, middle-to-middle, longest-to-longest in one ratio direction.

Solution walkthrough

01

Pair the corresponding sides

\[3<->6;~4<->8;~5<->10\]

The two side lists occupy matching positions, so each length in the second triangle is compared with the side in the same position in the first.

02

Test all three ratios in one direction

\[6/3=2;~8/4=2;~10/5=2\]

Using second-triangle length divided by first-triangle length keeps the comparison direction consistent.

03

Apply SSS similarity

\[\text{three}~\text{equal}~\text{corresponding}-\text{side}~\text{ratios}~->~\text{similar}~\text{triangles}\]

SSS similarity applies because every pair of corresponding sides is proportional with the same positive factor 2.

04

State and check the conclusion

\[(6,8,10)=2(3,4,5)\]

Doubling each first-triangle side reproduces the complete second-triangle side list, confirming similarity.

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Another way

  1. Factor the second side-length triple directly: (6,8,10)=2(3,4,5).

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Common mistake

Do not compare additive differences. Similarity requires one common multiplicative ratio for all corresponding sides.

Answer SSS comparison: T1 sides 3, 4, 5 correspond in order to T2 sides 6, 8, 10. The ratios are 6/3=2, 8/4=2, and 10/5=2. One common factor relates all three sides, so the triangles are similar by SSS.
All three sorted side ratios decide SSS similarity.