Q. 332 - Intermediate Algebra

Question

A goat enclosure is in the shape of a right triangle. One leg of the enclosure is built against the side of the barn. The other leg is $4$ feet more than the leg against the barn. The hypotenuses is $8$ feet more than the leg along the barn. Find the three sides of the goat enclosure.

Step-by-Step Solution

Answer

The three sides of the goat enclosure are $12 feet, 20 feet, 16 feet$.

Step 1. Assumption of integers.

Assume one side of the building to be $x feet$, hypotenuse to be $x+8 feet$ and other side to be $x+4 feet$.

We know, the Pythagoras theorem,

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

Substitute the values in the equation of Pythagoras theorem,

$$\begin{gathered} x^2+{\left(x+4\right)}^2={\left(x+8\right)}^2 ...... \left(i\right) \\ {x}^2+x^2+16+8x=x^2+64+16x \\ {x}^2-8x-48=0 \end{gathered}$$

Step 2. Factor the equation.

On factoring,

$$\begin{gathered} x^2-12x+4x-48=0 \\ x\left(x-12\right)+4\left(x-12\right)=0 \\ \left(x-12\right)\left(x+4\right)=0 \end{gathered}$$

Use the zero product property,

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

Then,

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

As we know, one side cannot be negative, so $x=12$.

Then the length of the hypotenuse,

$$\begin{gathered} =x+8 \\ =12+8 \\ =20 \end{gathered}$$

Also, the length of the other side,

$$\begin{gathered} =x+4 \\ =12+4 \\ =16 \end{gathered}$$

Step 3. Check the answers.

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

$$\begin{gathered} {12}^2+{16}^2={20}^2 \\ 144+256=400 \\ 400=400 \end{gathered}$$

This is true.

Hence, the three sides are $12 feet, 20 feet, 16 feet$.