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In triangle ABC, AD is the bisector of angle A (D on BC). Which statement correctly gives the ratio of the areas of triangle ABD to triangle ACD, and why?

1. Area(ABD)/Area(ACD) = AB/AC, because both triangles have the same height from A and BD/DC = AB/AC
2. Area(ABD)/Area(ACD) = AC/AB, because the bisector splits BC inversely to the adjacent sides
3. Area(ABD)/Area(ACD) = (AB/AC)², by similarity of the two sub-triangles
4. Area(ABD)/Area(ACD) = 1, because an angle bisector always halves the area

Correct answer: Area(ABD)/Area(ACD) = AB/AC, because both triangles have the same height from A and BD/DC = AB/AC

Solution

Triangles ABD and ACD share the same vertex A and their bases BD, DC lie on the same line BC, so both have the same height h (the perpendicular distance from A to BC). Therefore Area(ABD)/Area(ACD) = (1/2 * BD * h)/(1/2 * DC * h) = BD/DC. By the angle bisector theorem, BD/DC = AB/AC. Hence Area(ABD)/Area(ACD) = AB/AC.

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