Question

Geometry The hypotenuse of a right triangle has length $20\text{cm}$. The sum of the lengths of the other two sides is $28\text{cm}$. Find the lengths of the other two sides of the triangle.

Short answer

The length of the other two sides of triangles are$16\text{cm}$ and $12\text{cm}$.

Step-by-step solution

Step 1. Write the given information.

The length of the hypotenuse of the right triangle is$20\text{cm.}$

The sum of the length of the other two sides is $28\text{cm}$.

Step 2. Form the system of equations.

Let the other two sides of the triangle be x and y. So,

$\begin{array}{c}x+y=28\\ y=28-x\end{array}$

Now consider the expression.

$\begin{array}{c}\left(x+y\right)=28\\ {\left(x+y\right)}^2={28}^2\\ x^2+y^2+2xy=784\\ x^2+y^2=784-2x\left(28-x\right)\end{array}$

The expression for the hypotenuse in a right triangle is given by,

${\left(\text{Hypotenuse}\right)}^2={\left(\text{Base}\right)}^2+{\left(\text{Perpendicular}\right)}^2\mathrm{...}\left(1\right)$

Step 3. Form the equation and solve for x.

Substitute the values in equation (1) and solve for x.

$\begin{array}{c}{20}^2=784-2x\left(28-x\right)\\ 784-400-56x+2x^2=0\\ 2x^2-56x+384=0\\ x^2-28x+192=0\end{array}$

Factorize the above expression.

$\begin{array}{c}{x}^2-16x-12x+192=0\\ \left(x-16\right)\left(x-12\right)=0\end{array}$

Apply zero product property,

$\begin{array}{c}x-16=0\\ x=16\end{array}$

And,

$\begin{array}{c}x-12=0\\ x=12\end{array}$

So, if one side of the triangle is$16\text{cm}$ then the other is $12\text{cm}$.

Thus, the length of the other two sides of triangles are$16\text{cm}$ and $12\text{cm}$.