The lengths of the three sides of a triangle are given as 3 cm, 4 cm and 5 cm. The ratio of their corresponding altitudes to the opposite vertices will be:
Show answer & explanation
Correct answer: C
Explanation
The correct answer is option 3 i.e. 20∶15∶12.
In a triangle,
Area = (1/2) × base × corresponding height
Area using Heron’s formula:
s = (3 + 4 + 5)/2 = 6
Area = √[6(6−3)(6−4)(6−5)] = √(6×3×2×1) = √36 = 6 cm²
Now,
Height corresponding to 3 cm side = (2 × Area) / 3 = 12 / 3 = 4 cm
Height corresponding to 4 cm side = 12 / 4 = 3 cm
Height corresponding to 5 cm side = 12 / 5 = 2.4 cm
So, ratio = 4 : 3 : 2.4 = 40 : 30 : 24 = 10 : 7.5 : 6
Multiply by 2 to remove decimal:
20 : 15 : 12