The sides of a triangle are 35 cm, 54 cm and 61 cm respectively. The length of…

Concept

For a fixed triangle area A, the altitude h to a side of length b satisfies A = 1/2 × b × h, so h = 2A/b. Therefore, altitudes are inversely proportional to their corresponding side lengths, and the longest altitude is drawn to the shortest side.

When all three side lengths are known, Heron’s formula gives A = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2 is the semi-perimeter.

Application

1. The shortest side is 35 cm, so the longest altitude uses 35 cm as its base.

2. The semi-perimeter is s = (35 + 54 + 61)/2 = 75 cm.

3. The area is A = √[75(75 − 35)(75 − 54)(75 − 61)] = √[75 × 40 × 21 × 14] = √882000 = √(4202 × 5) = 420√5 cm2.

4. Using the 35 cm base, h = 2A/35 = (2 × 420√5)/35 = 24√5 cm.

Cross-check

The other altitudes are (2 × 420√5)/54 = 140√5/9 cm and (2 × 420√5)/61 = 840√5/61 cm. Both are smaller because their corresponding sides are longer than 35 cm.

Result

The longest altitude is 24√5 cm.