Question

The perimeter of a triangle ABC is 37 cm and the ratio between the lengths of its altitudes be 6: 5: 4. Find the lengths of its sides.
Let the sides be x cm, y cm, and (37 - x - y) cm. Also, let the lengths of altitudes be 6a cm, 5a cm, and 4a cm.

Answer 1

Consider that the sides be x cm, y cm, and (37 - x - y) cm. also, consider that the lengths of altitudes be 6a cm, 5a cm, and 4a cm.

∴ Area of a triangle = `1/2` x base x altitude

∴ `1/2 xx x xx 6a = 1/2 xx y xx 5a = 1/2 xx ( 37 - x - y) xx 4a`

6x = 5y = 148 - 4x - 4y

6x = 5y and 6x = 148 - 4x - 4y

6x - 5y = 0 and 10x + 4y = 148

Solving both the equations, we have
X = 10 cm, y = 12 cm and ( 37 - x - y ) cm = 15cm.