Question

select the correct answer
janet is fencing a triangular portion of her yard, which has measurements as shown. she needs fencing for the two sides of the yard that do not include the house. she also needs fencing for the third side of the yard, except for the portion taken up by the house.
what is the approximate length of fencing she needs?
a. 241 ft
b. 341 ft
c. 273 ft
d. 441 ft

Explanation:

Step1: Identify the triangle type

The given triangle is a right - angled isosceles triangle (since two angles are \(45^{\circ}\)). In a \(45 - 45-90\) triangle, the legs are equal. Let the length of each of the non - hypotenuse sides (the two sides that are not the side we are calculating for the fence) be \(a = 100\) ft.

Step2: Use the Pythagorean theorem

The Pythagorean theorem for a right - triangle is \(c=\sqrt{a^{2}+b^{2}}\). In a \(45 - 45 - 90\) triangle \(a = b\). So, \(c=\sqrt{100^{2}+100^{2}}=\sqrt{10000 + 10000}=\sqrt{20000}\).
We can also use the formula for the hypotenuse of a \(45 - 45-90\) triangle \(c=a\sqrt{2}\) (where \(a\) is the length of the leg). Substituting \(a = 100\), we get \(c = 100\sqrt{2}\approx100\times1.414 = 141.4\) (this is the hypotenuse). The total length of the fencing is \(100+100 + 141.4=341.4\approx341\) ft.

Answer:

B. 341 ft