Question

the length of the sides of a triangle from a G.P. if the perimeter of the triangle is 37 cm and the shortest side is of length 9 cm, find the length of the other two sides.

Answer 1

Given that the sides of the triangle are in G.P.

Let the sides of the triangle be $a,ar,a{r}^2$.

As we know that the perimeter of a triangle is equal to the sum of all sides of triangle.

Therefore,

$a+ar+a{r}^2=37.....\left(1\right)$

Given that the length of the shortest side is $9cm$.

Therefore,

$a=9$

Substituting the value of $a$ in ${eq}^n\left(1\right)$, we have

$9+9r+9r^2=37$

$\Rightarrow 9r^2+9r-28=0$

$\Rightarrow 9r^2+21r-12r-28=0$

$\Rightarrow 3r(3r+7)-4(3r+7)=0$

$\Rightarrow (3r+7)(3r-4)=0$

$\Rightarrow r=-\frac{7}{3}\text{or}r=\frac{4}{3}$

Since sides of triangle cannot be negative.

$\Rightarrow r\ne -ve$

$\therefore r=\frac{4}{3}$

Thus, sides of triangle are $9cm,12cm$ and $16cm$.