Q 6.107 - Intermediate Algebra

Question

Justine wants to put a deck in the corner of her backyard in the shape of a right triangle. The length of one side of the deck is $7$ feet more than the other side. The hypotenuse is $13$. Find the lengths of the two sides of the deck.

Step-by-Step Solution

Answer

The lengths of the two sides of the deck are $5 and 12 feet.$

Step 1. Given Information

There is a backyard of Justine where she wants to put a deck in the shape of a right triangle and the length of one side of the deck is $7$ feet more than the other side and the hypotenuse is $13.$

We have to find the lengths of the two sides of the deck.

Step 2. Assume the lengths of the two sides of the deck

Let the length of one side of the deck be x

So, the length of the other side of the deck will be $x+7.$

Step 3. Summarize into an equation

As we know the deck is in the shape of a right triangle, we can use the Pythagoras theorem.

Pythagoras theorem is

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

Substitute the values in the variables

$$\begin{gathered} x^2+{(x+7)}^2={13}^2 \\ {x}^2+x^2+49+14x=169 \\ 2x^2+14x=169-49 \\ 2x^2+14x=120 \end{gathered}$$

Step 4. Solve the equation

The equation we get is

$$\begin{gathered} 2x^2+14x=120 \\ 2x^2+14x-120=0 \\ 2(x^2+7x-60)=0 \\ 2(x^2+12x-5x-60)=0 \\ 2\left[x\right(x+12)-5(x+12\left)\right]=0 \\ 2(x-5)(x+12)=0 \end{gathered}$$

Step 5. Use the Zero Product Property

As we know $2\ne 0$

And $$\begin{gathered} x-5=0 \\ x=5 \end{gathered}$$

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

Step 6. Substitute the value of x in assumed lengths of the deck to find the exact lengths of the deck

Since x is the side of the triangle, it does not be negative therefore, we will eliminate the value $-12.$

If $x=5$

So, the length of one side of the deck will be $5$

and the length of the other side of the deck will be $5+7=12.$

Step 7. Verify the lengths of the side of the right triangle

Let's verify the sides of the right triangle by Pythagoras theorem

So, $$\begin{gathered} 5^2+{12}^2={13}^2 \\ 25+144=169 \\ 169=169 \end{gathered}$$

Thus, the sides of the deck are $5, 12 and 13 feet.$