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 the triangle.

Answer 1

Step 1: Find the length of each sides of the triangle

Given, perimeter of a triangle = $50cm$

Let the smaller side be $x.$

According to question,

So , the remaining sides are $2x-6cm$ and $x+4cm.$

Given , Perimeter of the triangle = $50cm.$

$$\begin{gathered} x+2x-6+x+4=50 \\ 4x-2=50 \\ 4x=50+2 \\ 4x=52 \\ x=\frac{52}{4} \\ x=13 \end{gathered}$$

So , the required sides of the triangle are

$$\begin{gathered} x=13cm \\ x+4=13+4=17cm \\ 2x-6=2\times 13-6=26-6=20cm \end{gathered}$$

Step 2: Use Heron's formula to find the area of triangle

Area of the triangle $=\sqrt{s\left(s-a\right)\left(s-b\right)(s-c)}$ (where ,$a,b,c$ are sides of triangle and $s$is semi perimeter)

Here, $\mathrm{a}=13\mathrm{cm},\mathrm{b}=17\mathrm{cm},\mathrm{c}=20\mathrm{cm}$

$$\begin{gathered} s=\frac{a+b+c}{2} \\ =\frac{13+17+20}{2} \\ =\frac{50}{2} \\ =25cm \end{gathered}$$

Area of the triangle $=\sqrt{25\times \left(25-13\right)\times \left(25-17\right)\times \left(25-20\right)}$

$$\begin{gathered} =\sqrt{25\times 12\times 8\times 5} \\ =\sqrt{5\times 5\times 2\times 2\times 3\times 2\times 2\times 2\times 5} \\ =5\times 2\times 2\sqrt{3\times 2\times 5} \\ =20\sqrt{30} \end{gathered}$$$$

Hence , Area of the triangle is $20\sqrt{30}c{m}^2.$