Find the area of the largest rectangle that can register in the triangle sides 5, 12 and 13 cm if 1 side of

This rectangle is on the base of the triangle
answer area 15 cm ^ 2

Let x = side on base of 5 cm
Let 12-y = side on height of 12 cm
By similar triangle 12/5 = y/x
x = 5y/12
A = 5y/12*(12-y) = 5y-5y^2/12
dA/dy = 5 - 10y/12 = 0 for maximum
60 = 10y
y = 6 cm
Area = 30 - 5*36/12 = 15 cm^2

this is a right angled triangle .let the base be the side with 5cm
y=x*(5-x)(12/5)=12x-12x^2/5
y'=12-24x/5

y'=0 at 12-24x/5=0
y'=0 at x=5/2

answer:=5/2*5/2(12/5)=15

A = (5 - x)*(12/5)x = 12x - (12/5)x²
dA/dx = 12 - (24/5)x = 0 → x = 5/2

Amax = 18 cm²