Q. 330 - Intermediate Algebra

Question

A stained glass window is shaped like a right triangle. The hypotenuse is $15 feet$. One leg is three more than the other. Find the lengths of the legs.

Step-by-Step Solution

Answer

The two required lengths of the legs of the stained glass window are $9 feet, 12 feet$.

Step 1. Assumption of integers.

Assume the length of the other leg to be $x$ and length of one leg to be $x+3 feet$.

We know, the Pythagoras theorem,

$a^2+b^2=c^2$

Substitute the values in the equation of Pythagoras theorem,

$$\begin{gathered} x^2+{\left(x+3\right)}^2={\left(15\right)}^2 ...... \left(i\right) \\ {x}^2+x^2+9+6x=225 \\ 2x^2+6x-216=0 \\ {x}^2+3x-108=0 \end{gathered}$$

Step 2. Factor the equation.

On factoring,

$$\begin{gathered} x^2+12x-9x-108=0 \\ x\left(x+12\right)-9\left(x+12\right)=0 \\ \left(x+12\right)\left(x-9\right)=0 \end{gathered}$$

Use the zero product property,

$$\begin{gathered} x+12=0 \\ x=-12 \end{gathered}$$

Then,

$$\begin{gathered} x-9=0 \\ x=9 \end{gathered}$$

As we know, one length of the leg cannot be negative, so $x=9$.

Then the length of the other leg,

$$\begin{gathered} =x+3 \\ =9+3 \\ =12 \end{gathered}$$

Step 3. Check the answers.

Substitute the length of the legs in equation (i),

$$\begin{gathered} 9^2+{12}^2={15}^2 \\ 81+144=225 \\ 225=225 \end{gathered}$$

This is true.

Hence, the two required lengths of the legs of the stained glass window are $9 feet, 12 feet$.