Question

The perimeter of a triangle $ABC$ is $37cm$ and the ratio between the lengths of its altitudes be $6:5:4$. Find the smallest length of its sides.

Answer 1

perimeter of a triangle ABC is 37cm.
Let the sides be x cm,y cm and Z cm. Also, let the lengths of altitudes be 6a cm,5a cm and4a cm.
Area=
$1/2\ast 6a\ast x=1/2\ast y\ast 5a$
$6x=5y$
like wise
$1/2\ast z\ast 4a=1/2\ast y\ast 5a$
$4z=5y$
So, 6x=5y=4z
As perimeter of a triangle ABC is 37cm.
$x+y+z=37$
$5/6x+y+5/4z=37$
$10y+12y+15y=37\ast 12$
$37y=37\ast 12$
$Y=12$
$x=12\ast 5/6=10$
$z=12\ast 5/4=15$
lengths of its sides x is 10cm,y is 12cm and z is 15cm.