Q. 331 - Intermediate Algebra

Question

A reflecting pool is shaped like a right triangle, with one leg along the wall of a building. The hypotenuse is $9 feet$ longer than the side along the building. The third side is $7 feet$ longer than the side along the building. Find the lengths of all three sides of the reflecting pool.

Step-by-Step Solution

Answer

The lengths of all the three sides of the reflecting pool are $8 feet, 15 feet, 17 feet$.

Step 1. Assumption of integers.

Assume the length of the side along the building to be $x feet$, hypotenuse to be $x+9 feet$ and third side to be $x+7 feet$.

We know, the Pythagoras theorem,

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

Substitute the values in the equation of Pythagoras theorem,

$$\begin{gathered} x^2+{\left(x+7\right)}^2={\left(x+9\right)}^2 ...... \left(i\right) \\ {x}^2+x^2+49+14x=x^2+81+18x \\ {x}^2-4x-32=0 \end{gathered}$$

Step 2. Factor the equation.

On factoring,

$$\begin{gathered} x^2-8x+4x-32=0 \\ x\left(x-8\right)+4\left(x-8\right)=0 \\ \left(x-8\right)\left(x+4\right)=0 \end{gathered}$$

Use the zero product property,

$$\begin{gathered} x-8=0 \\ x=8 \end{gathered}$$

Then,

$$\begin{gathered} x+4=0 \\ x=-4 \end{gathered}$$

As we know, one side of the triangle cannot be negative, so $x=8$.

Then the length of the hypotenuse,

$$\begin{gathered} =x+9 \\ =8+9 \\ =17 \end{gathered}$$

Also, the length of the third side,

$$\begin{gathered} =x+7 \\ =8+7 \\ =15 \end{gathered}$$

Step 3. Check the answers.

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

$$\begin{gathered} 8^2+{15}^2={17}^2 \\ 64+225=289 \\ 289=289 \end{gathered}$$

This is true.

Hence, the lengths of the three sides are $8 feet, 15 feet, 17 feet$.