How it works
Two figures are similar when their corresponding angles are equal and their corresponding sides are all in the same ratio k, the scale factor. For triangles, two matching angles are enough (AA similarity). Figures are congruent when k = 1 (for triangles: SSS, SAS, ASA, AAS, or HL for right triangles).
lengths × k · areas × k² · volumes × k³
Perimeters are lengths, so they scale by k. Areas scale by k², which is the step students most often forget. Shadow problems are similar triangles in disguise: at the same moment, every object's height-to-shadow ratio is the same.
Similarity And Congruence practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
Two similar triangles have side ratio 3:4. The smaller triangle has perimeter 36. Find the perimeter of the larger triangle.
Show answer and solution
Answer: 48
Perimeter scales like length, by 4/3. So the larger perimeter is 36 × 4 ÷ 3 = 48.
A 5 foot person casts a 6 foot shadow. At the same time a building casts a 42 foot shadow. Find the building height in feet.
Show answer and solution
Answer: 35 feet
Height ÷ shadow is the same for both: h ÷ 42 = 5 ÷ 6, so h = 42 × 5 ÷ 6 = 35.
Two similar triangles have side ratio 4:5. The smaller triangle has area 64. Find the area of the larger triangle.
Show answer and solution
Answer: 100
Areas scale by the square of the ratio, 16:25. So the larger area is 64 × 25 ÷ 16 = 100.
Similar triangles, scale factor 4. Small area 64. Large area?
Show answer and solution
Answer: 1024
Area scales by 4² = 16, so the large area is 64 × 16 = 1024.
Keep going
Related topics
Related practice sets in the app
All Geometry topics · Contest math practice by topic · MATHCOUNTS practice · AMC 8 practice