Question

The perimeter of a right angled triangle is $60cm$ and its hypotenuse is $25cm.$ Find the area of the triangle.

Answer 1

Step 1: Find the sides of the triangle

Let the other two sides i.e. base and perpendicular of a right angled triangle be $a$ and $b$respectively.

Given , Perimeter of right angled triangle = $60cm$

$$\begin{gathered} a+b+25=60 \\ a+b=60-25 \\ a+b=35 \\ b=35-a \end{gathered}$$

We know that in a right angled triangle ,

$$\begin{gathered} {\left(hypotenuse\right)}^2={\left(base\right)}^2+{\left(perpendicular\right)}^2 \\ {\left(25\right)}^2={\left(a\right)}^2+{\left(35-a\right)}^2 \\ 625=a^2+1225+a^2-70a \\ 2a^2-70a+600=0 \\ {a}^2-35a+300=0 \\ {a}^2-15a-20a+300=0 \\ a\left(a-15\right)-20\left(a-15\right)=0 \\ \left(a-15\right)\left(a-20\right)=0 \\ a-15=0,a-20=0 \\ a=15,a=20 \end{gathered}$$

Two cases arises in this case for two values of $a$

$$\begin{gathered} a=15,b=35-15=20 \\ a=20,b=35-20=15 \end{gathered}$$

So , the remaining two sides are $15cm$ and $20cm.$

Step 2: Find the area of the triangle

Area of the triangle $=\frac{1}{2}\times base\times height$

$$\begin{gathered} =\frac{1}{2}\times 20\times 15 \\ =10\times 15 \\ =150{\mathrm{cm}}^2 \end{gathered}$$

Hence, the area of a triangle is $150c{m}^2$.