Question

The altitude of a triangle is $6$ cm greater than its base. If its area is $108$ cm${}^2$. Find its base and height.

Answer 1

Let the base of the triangle be $x$.

It is given that the altitude of the triangle is $6$ cm greater than its base, therefore, the altitude of the triangle is $(x+6)$.

It is also given that the area of the triangle is $108$ cm${}^2$.

We know that the area of the triangle is $A=\frac{1}{2}\times b\times a$, therefore,

$108=\frac{1}{2}\left(x\right)(x+6)\Rightarrow x^2+6x=108\times 2\Rightarrow x^2+6x=216\Rightarrow x^2+6x-216=0\Rightarrow x^2+18x-12x-216=0\Rightarrow x(x+18)-12(x+18)=0\Rightarrow (x-12)=0,(x+18)=0\Rightarrow x=12,x=-18$

Since the length of the triangle cannot be negative thus, $x=12$.

Hence, the base of the triangle is $12$ cm.