About how many feet of fencing are needed to enclose a rectangular garden with a 12-foot-long side and a 26-foot-long diagonal? Round your answer to the nearest whole foot.

Answer 1

To solve this problem, we need to find the dimensions of the rectangular garden given one side and one diagonal.

We are given:

• One side of the rectangle ($a$) = 12 feet
• Diagonal of the rectangle ($d$) = 26 feet

According to the Pythagorean theorem, in a right triangle formed by the length, width, and diagonal of a rectangle: $$d^2=a^2+b^2$$ where $b$ is the other side of the rectangle.

Substituting the known values into this equation: $${26}^2={12}^2+b^2$$ Calculating ${26}^2$ and ${12}^2$: $$676=144+b^2$$ Now, we will isolate $b^2$: $$b^2=676-144=532$$ Taking the square root of both sides to find $b$: $$b=\sqrt{532}\approx 23.065$$

Now we can find the perimeter $P$ of the rectangle, which is given by the formula: $$P=2(a+b)$$ Substituting the known values: $$P=2(12+\sqrt{532})=2(12+23.065)\approx 2(35.065)\approx 70.13$$

Rounding to the nearest whole foot, the total amount of fencing needed is approximately: $$\boxed{{\displaystyle 70}}\text{ feet}$$