In a triangle ABC, AD is the altitude from A to BC. If AD = 12 cm, AB = 13 cm, and AC = 14 cm, find the length of BC.
Using Pythagoras' theorem on triangles ABD and ACD: Let BD = x and DC = y, so BC = x + y. From triangle ABD: AB² = AD² + BD² 13² = 12² + x² 169 = 144 + x² x² = 25 → x = 5 cm. From triangle ACD: AC² = AD² + DC² 14² = 12² + y² → 196 = 144 + y² y² = 52 → y = √52 ≈ 7.21 cm. BC = x + y = 5 + 7.21 = 12.21 cm.
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