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Math

Similar Triangles

5 min readMedium5-question drill

Two triangles are *similar* when they have the same shape but different size — corresponding angles match and corresponding sides are in the same ratio. The SAT uses similarity to test proportional reasoning in geometry.

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Linear, area, volume ratios on similar figures
Side ratio (linear)Area ratioVolume ratio
1 : 21 : 41 : 8
1 : 31 : 91 : 27
2 : 54 : 258 : 125
1 : k1 : $k^2$1 : $k^3$
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Recognizing similarity in a triangle problem
Are two angles of one triangle equal to two angles of the other?
Yes ↓
Triangles are similar by AA — set up the ratio of corresponding sides
No ↓
Is there a line drawn parallel to one side of a triangle?
Yes ↓
The smaller triangle is similar to the larger by AA (same angle at the shared vertex + parallel lines = same angles)
No ↓
Are all three pairs of sides in the same ratio?
Yes ↓
Triangles are similar by SSS
No ↓
Need more info — check for SAS (two sides proportional + included angle)
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Quick check

Set up the proportion using corresponding sides (read the similarity statement carefully). Remember to SQUARE the linear ratio if the question is about area.

Two similar triangles have corresponding sides in the ratio 1:2. If a side of the smaller triangle is 8, what is the corresponding side of the larger triangle?

Worked examples

Example 1
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Example 2
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Common pitfalls

Using the linear ratio for area

If sides scale by 3, area scales by 9 (not 3). The most common SAT trap on similar-triangle area problems. Square the side ratio for area. Cube it for volume.

Setting up the proportion with mismatched corresponding sides
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Forgetting that AA only requires two angles

If two pairs of angles are equal, the third pair MUST also be equal (angles sum to 180° in every triangle). So AA is enough to prove similarity — you don't need a third angle or any side.

Mixing up similar with congruent

Similar = same shape, different size (scale factor can be anything). Congruent = same shape, same size (scale factor = 1). SAT problems usually want similar — corresponding sides PROPORTIONAL, not equal.

Key takeaways

  • Similar triangles: same angles, proportional sides. Scale factor kk = side ratio.

  • AA, SSS, SAS — any one establishes similarity.

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  • A line parallel to one side of a triangle creates a similar smaller triangle.

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Try it yourself

5 practice questions on Similar Triangles, drawn from the question bank. The tutor is one click away if you get stuck.

Lesson v1 · generated 5/2/2026 · the floating tutor knows you're on this lesson — ask anything.