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. 441 ft
c. 341 ft
d. 273 ft

Explanation:

Step1: Identify Triangle Type

The triangle is a right - isosceles triangle (two 45° angles and a right angle), so the two equal sides (excluding the side with the house) are 100 ft each? Wait, no, the hypotenuse? Wait, no, in a right - isosceles triangle, the legs are equal, and the hypotenuse \(c = l\sqrt{2}\), where \(l\) is the length of a leg. Wait, the triangle has a right angle and two 45° angles, so it's an isosceles right triangle. The two sides (the non - house sides) are 100 ft? Wait, no, the diagram shows a right triangle with two 45° angles, and one of the legs (the side not adjacent to the house) is 100 ft? Wait, no, let's re - examine. The house is along a side, and we have a right - isosceles triangle. The two equal sides (the ones with the 45° angles) are the legs, and the hypotenuse is the side opposite the right angle. Wait, the length of the hypotenuse of an isosceles right triangle with leg length \(a\) is \(a\sqrt{2}\). But here, we need to find the total fencing. Wait, the two sides of the yard (the ones with the 45° angles) are each 100 ft? No, wait, the problem says "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." Wait, the triangle is a right - isosceles triangle. Let's denote the legs as \(a\) and \(b\), and the hypotenuse as \(c\). Since it's a right - isosceles triangle, \(a = b\), and the angles are 45°, 45°, 90°. The length of the hypotenuse \(c=a\sqrt{2}\). Wait, but in the diagram, one of the legs (the side not covered by the house) is 100 ft? Wait, no, the house is 100 ft long, and we need to find the length of the third side (the hypotenuse - like side) minus the 100 ft covered by the house? Wait, no, let's calculate the length of the hypotenuse first. Wait, in a right - isosceles triangle, if the legs are of length \(x\), then the hypotenuse is \(x\sqrt{2}\). But here, maybe the legs are 100 ft? Wait, no, the two sides with the 45° angles are the legs, and we need to find the length of the third side (the base) minus the 100 ft (the house length). Wait, no, let's do the math. The two equal sides (the legs) are each 100 ft? No, wait, the length of the hypotenuse of an isosceles right triangle with leg length \(a\) is \(a\sqrt{2}\). If \(a = 100\) ft, then the hypotenuse is \(100\sqrt{2}\approx141.42\) ft. But that can't be. Wait, maybe the legs are longer. Wait, no, the problem is about fencing. Let's think again. The total fencing needed is the sum of the two 100 - ft sides (the legs) plus the length of the hypotenuse minus the 100 - ft portion covered by the house. Wait, the hypotenuse of an isosceles right triangle with leg length \(a\) is \(a\sqrt{2}\). Wait, if the legs are, say, \(x\), then the hypotenuse is \(x\sqrt{2}\). But in the diagram, the house is 100 ft, so we need to find the length of the hypotenuse, then subtract 100 ft, and add the two legs. Wait, no, the two legs are each 100 ft? No, that doesn't make sense. Wait, let's calculate the length of the hypotenuse first. Wait, in a right - isosceles triangle, the hypotenuse \(c = a\sqrt{2}\), where \(a\) is the length of a leg. If we assume that the legs are 100 ft, then the hypotenuse is \(100\sqrt{2}\approx141.42\) ft. But that's not matching the options. Wait, maybe the legs are not 100 ft. Wait, maybe the two sides with the 45° angles are the legs, and their length is \(x\), and the hypotenuse is \(x\sqrt{2}\). But we need to find the total fencing: the two legs (each \(x\)) plus the hypotenuse minus the 100 ft (the house). Wait, but the options are 241, 441, 341, 273. Let's try \(x = 100\) ft. Then the hypotenuse is \(100\sqrt{2}\approx141.42\) ft. Then the total fencing would be \(100 + 100+(141.42 - 100)=100 + 100+41.42 = 241.42\approx241\) ft? No, that's option A. Wait, but let's check again. Wait, maybe the legs are not 100 ft. Wait, the triangle is a right - isosceles triangle, so the two legs are equal, and the hypotenuse is \(leg\times\sqrt{2}\). The house is along the hypotenuse? No, the house is along one of the legs? Wait, the diagram shows the house inside the right - isosceles triangle, with the base of the triangle (the hypotenuse) being the side with the house? No, the right angle is at the top, and the house is at the bottom. So the two legs are the sides with the 45° angles, and the hypotenuse is the bottom side (with the house). Wait, that makes more sense. So the two legs are each 100 ft? No, the length of each leg is \(x\), and the hypotenuse (the bottom side) is \(x\sqrt{2}\). The house is 100 ft long, so the length of the hypotenuse minus the house length is \(x\sqrt{2}-100\). And we need to add the two legs (\(x + x\)) plus (\(x\sqrt{2}-100\)). Wait, but we know that in the diagram, one of the legs is 100 ft? No, the problem says "the two sides of the yard that do not include the house" are the legs, each 100 ft? Wait, no, let's calculate the hypotenuse first. If the legs are 100 ft, then the hypotenuse is \(100\sqrt{2}\approx141.42\) ft. Then the total fencing is \(100 + 100+(141.42 - 100)=241.42\approx241\) ft, which is option A. Wait, but let's check the calculation again. Wait, maybe the legs are longer. Wait, no, the answer options include 241, which is close to our calculation. So the total fencing is the sum of the two legs (100 ft each) plus the length of the hypotenuse minus the 100 ft (the house). The hypotenuse of an isosceles right triangle with leg length 100 ft is \(100\sqrt{2}\approx141.42\) ft. So \(100 + 100+(141.42 - 100)=241.42\approx241\) ft.

Step2: Select the Answer

Based on the calculation, the approximate length of the fencing is 241 ft, which corresponds to option A.

Answer:

A. 241 ft