Question

The sides of a right-angled triangle containing the right angle are $5x$ cm and $3x - 1$ cm. If its area is $60c{m^2}$, find its perimeter.

Answer 1

Hint:
Use the given area and length of sides of the right triangle to form an equation. Solve the equation to find the value of $x$. Then, length of sides can be calculated by substituting the values of $x$ and applying Pythagoras theorem. And add all the sides to determine the perimeter of the triangle.

Complete step by step solution:
We will use the given length of sides of the triangle, $5x$ cm and $3x - 1$ cm to draw a corresponding diagram.

We know that the area of a triangle is given as $\dfrac{1}{2} \times {\text{base}} \times {\text{height}}$
The area of the given triangle is $60c{m^2}$
Then, $60 = \dfrac{1}{2} \times \left( {5x} \right) \times \left( {3x - 1} \right)$
Solve the above equation to find the value of $x$
$
  60 = \dfrac{1}{2} \times \left( {5x} \right) \times \left( {3x - 1} \right) \\
   \Rightarrow 120 = 15{x^2} - 5x \\
   \Rightarrow 15{x^2} - 5x - 120 = 0 \\
   \Rightarrow 3{x^2} - x - 24 = 0 \\
 $
Factorise the above equation and then equate each factor to 0,
$
   \Rightarrow 3{x^2} - 9x + 8x - 24 = 0 \\
   \Rightarrow 3x\left( {x - 3} \right) + 8\left( {x - 3} \right) = 0 \\
   \Rightarrow \left( {3x + 8} \right)\left( {x - 3} \right) = 0 \\
   \Rightarrow x = - \dfrac{8}{3},3 \\
$
But, the length of the side cannot be negative, therefore, $x = 3$.
Therefore, the two sides of the triangle are $5\left( 3 \right) = 15$cm and $3\left( 3 \right) - 1 = 8cm$
We can find the remaining side using the Pythagoras theorem.
Pythagoras theorem states that the square of Hypotenuse is equal to the sum of squares of other two sides.
Therefore,
$
   \Rightarrow A{C^2} = {\left( {15} \right)^2} + {\left( 8 \right)^2} \\
   \Rightarrow A{C^2} = 225 + 64 \\
   \Rightarrow A{C^2} = 289 \\
   \Rightarrow AC = 17 \\
$
The perimeter is the sum of all sides of a triangle.
$15 + 8 + 17 = 40cm$

Note:
Area is the space covered by a shape and perimeter is the length of the boundary of the shape. Also, in a right triangle, the longest side is the hypotenuse and is opposite the right angle.