Question

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.
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feet, feet, feet
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Explanation:

Step1: Define variables

Let the side along the building be $x$ feet. Then the hypotenuse is $(x + 9)$ feet and the third - side is $(x+7)$ feet.

Step2: Apply Pythagorean theorem

By the Pythagorean theorem $a^{2}+b^{2}=c^{2}$ (where $c$ is the hypotenuse and $a,b$ are the legs), we have $x^{2}+(x + 7)^{2}=(x + 9)^{2}$.
Expand the equations:
$x^{2}+x^{2}+14x + 49=x^{2}+18x + 81$.

Step3: Simplify the equation

Combine like - terms:
$x^{2}+x^{2}+14x + 49-(x^{2}+18x + 81)=0$.
$x^{2}-4x - 32 = 0$.

Step4: Solve the quadratic equation

Factor the quadratic equation: $(x - 8)(x+4)=0$.
Set each factor equal to zero: $x-8 = 0$ gives $x = 8$; $x + 4=0$ gives $x=-4$. Since length cannot be negative, we discard $x=-4$.

Step5: Find the lengths of the sides

The side along the building $x = 8$ feet.
The hypotenuse is $x + 9=8 + 9 = 17$ feet.
The third side is $x + 7=8 + 7 = 15$ feet.

Answer:

8 feet, 15 feet, 17 feet