Question

1. jessie is mowing her back yard that is in the shape of a right triangle. the shortest leg is 3 meters less than the second side, and the hypotenuse is 15 meters long. what are the lengths of the two sides?

Explanation:

Step1: Set up variables

Let the second - side length be \(x\) meters. Then the shortest leg is \((x - 3)\) meters.

Step2: Apply the Pythagorean theorem

By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (where \(a=x - 3\), \(b = x\), \(c = 15\)), we have \((x - 3)^{2}+x^{2}=15^{2}\).
Expand \((x - 3)^{2}\): \(x^{2}-6x + 9+x^{2}=225\).
Combine like terms: \(2x^{2}-6x+9 - 225=0\), so \(2x^{2}-6x - 216 = 0\).
Divide through by \(2\): \(x^{2}-3x - 108=0\).

Step3: Factor the quadratic equation

Factor \(x^{2}-3x - 108\): \((x - 12)(x + 9)=0\).
Set each factor equal to zero: \(x-12 = 0\) or \(x + 9=0\).
Solve for \(x\): \(x=12\) or \(x=-9\). Since length cannot be negative, \(x = 12\).

Step4: Find the length of the other side

If \(x = 12\), then the shortest leg \(x-3=12 - 3=9\).

Answer:

The lengths of the two sides are \(9\) meters and \(12\) meters.