Numerical answer
ABCD is a square of side 12 cm. E is the midpoint of side BC. The line segments AE and BD intersect at F. Find the area (in cm²) of triangle BEF.
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Answer: 12
Place A = (0, 0), B = (12, 0), C = (12, 12), D = (0, 12). Then E = (12, 6).
AE: points (12t, 6t). BD: points (12 − 12s, 12s). Equating: 12t = 12 − 12s and 6t = 12s → t = 2s → 24s = 12 − 12s → s = 1/3, t = 2/3. So F = (8, 4).
Triangle BEF has vertical side BE = 6 (from (12, 0) to (12, 6)) and F is 12 − 8 = 4 units away horizontally.
Area = ½ × 6 × 4 = 12 cm².
Shortcut: Triangles FBE and FDA are similar with ratio BE : DA = 1 : 2, so F divides BD in ratio 1 : 2; its distance from BC is ⅓ × 12 = 4.
AE: points (12t, 6t). BD: points (12 − 12s, 12s). Equating: 12t = 12 − 12s and 6t = 12s → t = 2s → 24s = 12 − 12s → s = 1/3, t = 2/3. So F = (8, 4).
Triangle BEF has vertical side BE = 6 (from (12, 0) to (12, 6)) and F is 12 − 8 = 4 units away horizontally.
Area = ½ × 6 × 4 = 12 cm².
Shortcut: Triangles FBE and FDA are similar with ratio BE : DA = 1 : 2, so F divides BD in ratio 1 : 2; its distance from BC is ⅓ × 12 = 4.
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