Find the number of triangles in the given figure.
2024
Find the number of triangles in the given figure.
- A.
36
- B.
40
- C.
44
- D.
48
Attempted by 1 students.
Show answer & explanation
Correct answer: D
Concept: To count every triangle in a compound figure, list them by how many of the smallest (simplest) regions each one is built from — 1, 2, 3, and so on — checking each combination truly has three straight sides meeting at three vertices. Sum the counts across every size-class; stopping after only the smallest visible triangles undercounts the figure.
Application: Label the figure as shown — outer square ABCD, edge points E, F, G, H, I, J, K, L, M, N, O, P, and interior grid points Q, R, S, T, U, V, W, X, Y with U at the centre. Counting every distinct triangle by the number of simplest regions each is made of:
Made from how many regions | Example triangles | Count |
|---|---|---|
Made from 1 region (simplest) | APQ, AEQ, QTU, QRU, BGS, BHS, RSU, SUV, TUW, UWX, NWD, WDM, UVY, UXY, JCY, YKC | 16 |
Made from 2 regions | QUW, QSU, SYU, UWY | 4 |
Made from 3 regions | AOU, AFU, FBU, BIU, UIC, ULC, ULD, OUD | 8 |
Made from 4 regions | QYW, QSW, QSY, SYW | 4 |
Made from 6 regions | AUD, ABU, BUC, DUC | 4 |
Made from 7 regions | QMC, ANY, EBW, PSD, CQH, AGY, DSK, BJW | 8 |
Made from 12 regions | ABD, ABC, BCD, ACD | 4 |
Cross-check: Adding every size-class — 16 + 4 + 8 + 4 + 4 + 8 + 4 — gives 48 triangles in total, with no size-class double-counted or skipped.