Question

The altitude and the median from the same vertex of a triangle trisect the angle. Find the area of the triangle, if the length of the shorter side is 4√ 3 in.

Answer 1

24√3 in²

Explanation:

The triangle in question is a 30°-60°-90° right triangle. Such triangles have side lengths in the ratio ...

1 : √3 : 2

Given that the short side has length 4√3, the side at right angles to it will have length (4√3)(√3) = 12. Then the area is ...

A = 1/2bh = 1/2(4√3)(12) = 24√3 . . . square inches

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If you make the assumption that the triangle is a right triangle, and the angle being trisected is the right angle, then you find that assumption is true. If you work through the algebra and geometry, you can prove this is the only possible such triangle. That proof is beyond the scope of this answer.