Question

a building lot for sale in the city is in the shape of a right triangle, as shown to the right. the side not bordering a street is 24 ft long. one of the other sides is 4 ft longer than the remaining side. find the lengths of the sides. the length of the smaller side is □ and the length of the larger side is □

Explanation:

Step1: Set up variables

Let the length of the shorter side be \(x\) ft. Then the length of the longer side is \((x + 4)\) ft.

Step2: Apply the Pythagorean theorem

Since it is a right - triangle, by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). Here, \(a=x\), \(b = 24\), \(c=x + 4\). So \(x^{2}+24^{2}=(x + 4)^{2}\).
Expand \((x + 4)^{2}\) using the formula \((a + b)^{2}=a^{2}+2ab+b^{2}\), we get \(x^{2}+24^{2}=x^{2}+8x + 16\).
Subtract \(x^{2}\) from both sides: \(24^{2}=8x + 16\).
Calculate \(24^{2}=576\), so \(576=8x + 16\).
Subtract 16 from both sides: \(576-16=8x\), \(560 = 8x\).
Divide both sides by 8: \(x=\frac{560}{8}=70\).
The longer side is \(x + 4=70 + 4=74\).

Answer:

The length of the smaller side is \(70\) ft and the length of the larger side is \(74\) ft.