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Can the three altitudes of a triangle be in the ratio 2: 5: 6? State whether such a triangle can exist and why.

1. Yes, because any three positive numbers can serve as the altitudes of a triangle.
2. No, because the corresponding side lengths would violate the triangle inequality.
3. Yes, because the altitudes are always directly proportional to the sides.
4. No, because the altitudes of any triangle must all be equal.

Correct answer: No, because the corresponding side lengths would violate the triangle inequality.

Solution

Since area = (1/2)*base*height is the same for each side, the sides are inversely proportional to the altitudes. If altitudes are 2: 5: 6, the sides are proportional to 1/2: 1/5: 1/6 = 15: 6: 5 (multiplying by 30). Check triangle inequality on sides 15, 6, 5: 6 + 5 = 11 < 15, so the inequality fails. Therefore no such triangle exists.

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