Question

The length of the sides of a right triangle is $5x+2,5x$ and $3x-1$. Find the length of each side.

Answer 1

Step 1: Determine the quadratic equation in $x$

Let the sides of the triangle are $5x+2,5x$ and $3x-1$.

By the Pythagoras theorem

$$\begin{gathered} {\left(5x+2\right)}^2={\left(5x\right)}^2+{\left(3x-1\right)}^2 \\ \Rightarrow 25x^2+20x+4=25x^2+9x^2-6x+1 \\ \Rightarrow 9x^2-26x-3=0 \end{gathered}$$

Step 2: Determine the sides of the triangle

By factorization method

$$\begin{gathered} 9x^2-26x-3=0 \\ \Rightarrow 9x^2-27x+x-3=0 \\ \Rightarrow 9x\left(x-3\right)+1\left(x-3\right)=0 \\ \Rightarrow \left(x-3\right)\left(9x+1\right)=0 \\ \therefore x=\frac{-1}{9},3 \end{gathered}$$

The side of the triangle cannot be negative.

So, $x=3$

The sides of the triangle are $\left(5\times 3+2\right),\left(5\times 3\right),\left(3\times 3-1\right)=17,15,8$.

Hence, the length of the sides of the triangle are $17,15$ and $8$.