Question

The perimeter of a right angled triangle is $12cm$, if the hypotenuse is $5cm$, how do you find the area and two sides of a triangle?

Answer 1

Hint:Start by mentioning the definition of perimeter of a triangle. Then we will mention the formula for the perimeter of the triangle and substitute values in the formula. Then finally evaluate the conditions and solve for the dimensions of the triangle and area of the triangle.

Complete step by step answer:
First we will start off by mentioning the definition of perimeter. So, perimeter is a path that surrounds a two-dimensional shape. This term may be used either for the path, or its length in one dimension.
Now, consider the legs of the triangle as $x,y$.
Perimeter of a triangle is given by the sum of all the sides. Hence, we can write $x + y + 5 = 12$. If we further simplify we can write $y = 7 - x$.
Now if we apply the Pythagoras theorem in the given triangle, we can write, ${x^2} + {y^2} = {5^2}$.
Substitute $y = 7 - x$ in the equation ${x^2} + {y^2} = {5^2}$.
\[
{x^2} + {y^2} = {5^2} \\
{x^2} + {(7 - x)^2} = {5^2} \\
{x^2} + {x^2} - 14x + 49 = 25 \\
2{x^2} - 14x + 24 = 0 \\
{x^2} - 7x + 12 = 0 \\
(x - 3)(x - 4) = 0 \\
\]
Hence, the values of $x$ are $3,4$.
Now we will solve for the value of $y$.
$
y = 7 - x \\
y = 7 - 3 \\
y = 4 \\
$ and
$
y = 7 - x \\
y = 7 - 4 \\
y = 3 \\

$
Hence, the lengths of the legs of the triangle are $3,4\,cm$.
Now we will evaluate the area of the triangle.
Area of the triangle is given by $\dfrac{1}{2} \times b \times h$.
Now, we will substitute values in the formula and solve for the area of the triangle.
$
A = \dfrac{1}{2} \times b \times h \\
A = \dfrac{1}{2} \times 3 \times 4 \\
A = 6\,c{m^2} \\
$
Hence, the area of the triangle is $6\,c{m^2}$.

Note: While mentioning the definition, make sure to mention the terms properly. Reduce the terms by factorisation. While choosing any variable, for any unknown term, choose according to the conditions.