Question

The side of a right angled triangle containing the right angle are (3x)cm and (3x+2)cm. Calculate the perimeter of the triangle, if the area of the triangle is 24cm2​

Answer 1

Answer: 24 cm

Step-by-step explanation:

Let 3x be the length of the base of the triangle.

Let (3x + 2) be the height of the triangle.

The area is $A=24\,\rm{cm}^2$.

Calculate the are of the triangle.

$\begin{aligned}A&=\frac{1}{2}bh\\24&=\frac{1}{2}\times(3x)\times(3x+2)\\48&=9x^2+6x\\9x^2+6x-48&=0\end{aligned}$

Solve the above quadratic equation.

$\begin{aligned}9x^2+6x-48&=0\\3x^2+2x-16&=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\text{Divided the equation by 3})\\3x^2+8x-6x-16&=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\text{Splitting the middle term})\\x(3x+8)-2(3x+8)&=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\text{Taking common})\\(3x+8)(x-2)&=0\end{aligned}$

Thus, x can have the values 2 and -8/3.

Since the side of a triangle cannot be negative, the value of x is 2.

Hence,

Length of base is $3x=3\times2=6\,\rm{cm}$.

Height of the triangle is $3x+2=3\times2+2=8\,\rm{cm}$.

Calculate the length of the hypotenuse using Pythagoras theorem.

$\begin{aligned}\rm{Hypotenuse}&=\sqrt{6^2+8^2}\\&=\sqrt{36+64}\\&=\sqrt{100}\\&=10\,\rm{cm}\end{aligned}$

Calculate the perimeter.

$\begin{aligned}\rm{Perimeter}&=6+8+10\\&=24\,\rm{cm}\end{aligned}$