Question

The perimeter of a triangle is $50cm$. One side of a triangle is $4cm$ longer than the smaller side and the third side is $6cm$ less than twice the smaller side. Find the area of a triangle.

Answer 1

Hint: Here we are asked to find the area of the triangle by using the given data. We don’t have the direct values or length of the sides of a triangle instead we are given some statements using those statements we will form the expression for the length of the sides of a triangle. Then we will use the formula to find the area of a triangle.

Formula used:
The area of a triangle: $\sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} $
$s = \dfrac{{a + b + c}}{2}$
$s$ - semiperimeter
$a,b,c $- sides of the triangle
The perimeter of a triangle: $a + b + c$

Complete step by step answer:
It is given that the triangle’s perimeter is $50cm$. By the formula of the perimeter, we get
$a + b + c = 50$.
Let $a,b,c$ be the sides of a triangle and let $c$ be the smallest side of the triangle. It is given that one side of a triangle is four more than the smallest side. Let $a$ be that side so we get $a = c + 4$. And it is also given that another side is six less than twice the smallest side. Let $b$ be this side so we get $b = 2c - 6$.
Now we got the sides of the triangle as $a = c + 4$, $b = 2c - 6$ and $c$.
Noe let us find the value $c$ by using the perimeter. We know that $a + b + c = 50$ substituting the values of $a\& b$ in this we get $\left( {c + 4} \right) + \left( {2c - 6} \right) + c = 50$.
On simplifying this we get
$4c = 52$
$c = 13$
Substituting these in $a\& b$ we get
 $a = 13+4 = 17$and $b = 2 \times 13-6 = 20$
Now to find the area of the triangle we need to find the value of the term $s$ which is called semiperimeter.
$s = \dfrac{{a + b + c}}{2}$
$s = \dfrac{{50}}{2}$ (Since $a + b + c$ is the perimeter of the triangle)
$ \Rightarrow s = 25$
Thus, the triangle’s area will be $\sqrt {25\left( {25 - 17} \right)\left( {25 - 20} \right)\left( {25 - 13} \right)} $
On simplifying this we get
$ \Rightarrow \sqrt {25\left( 8 \right)\left( 5 \right)\left( {12} \right)} $
$ \Rightarrow \sqrt {2000} $
$ \Rightarrow 20\sqrt 5 $
Thus, we found that the area of the triangle is $20\sqrt 5 c{m^2}$.

Note:
The selection of the formula is important in this problem since we also have another formula to find the area of the triangle that is $\dfrac{1}{2}bh$ where $b - $the base of the triangle and $h - $perpendicular height of the triangle. Since in this problem we only have the sides of the triangle we have used the formula $\sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} $.