Congruent triangles are identical — same angles and same side lengths (established by SSS, SAS, ASA, AAS, or HL for right triangles). Similar triangles have the same shape but not necessarily the same size: their angles are equal and their corresponding sides are in a constant ratio. On the GRE, similarity is usually spotted by AA — if two angles match, the triangles are similar. Once similar, set up a proportion of corresponding sides to find a missing length. Two facts the GRE loves to test: the ratio of perimeters equals the ratio of sides, but the ratio of areas equals the square of the ratio of sides. Shadow and "distance across a river" problems are similar-triangle problems in disguise.
✅ Solved examples
1. Two similar triangles have corresponding sides in the ratio 3 : 5. A side of the smaller is 6. Find the corresponding side of the larger.
Set up the proportion 3/5 = 6/x → 3x = 30 → x = 10.
2. Two similar triangles have sides in the ratio 2 : 3. The perimeter of the smaller is 24. Find the perimeter of the larger.
Perimeters share the side ratio: 2/3 = 24/P → 2P = 72 → P = 36.
3. Two similar triangles have sides in the ratio 2 : 3. The area of the smaller is 8. Find the area of the larger.
Area ratio = (2/3)² = 4/9. So 4/9 = 8/A → 4A = 72 → A = 18.
4. A 6-foot-tall person casts a 4-foot shadow at the same time a tree casts a 20-foot shadow. Find the height of the tree.
The person and tree form similar right triangles: 6/4 = h/20 → 4h = 120 → h = 30 feet.
✏️ Practice — try these, take hints as needed
1. Two similar triangles have sides in the ratio 1 : 4. A side of the smaller is 5. Find the corresponding side of the larger.
Set up 1/4 = 5/x.
Cross-multiply.
x = 5 × 4.
20
2. Two similar triangles have sides in the ratio 3 : 4. The larger has a perimeter of 48. Find the smaller perimeter.
Perimeters share the ratio 3 : 4.
3/4 = P/48.
4P = 144.
36
3. The areas of two similar triangles are in the ratio 9 : 16. Find the ratio of their corresponding sides.
Side ratio is the square root of the area ratio.
√9 : √16.
3 : 4.
3 : 4
4. A 5-foot pole casts a 3-foot shadow while a building casts a 24-foot shadow. Find the height of the building.
Similar triangles: 5/3 = h/24.
Cross-multiply: 3h = 120.
Divide by 3.
40 feet
5. Two similar triangles have sides in the ratio 2 : 5. The smaller has an area of 12. Find the area of the larger.
Area ratio = (2/5)² = 4/25.
4/25 = 12/A.
4A = 300.
75
📝 Topic test — 8 questions
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