Question

joes roof forms a 45° angle with the top of the house. if the house is 120 ft wide, how much roof edge will need lights?
none of the other answer options are correct.
60√2 ft
60√3 ft
120√3 ft
120√2 ft

Explanation:

Step1: Analyze the right - triangle formed

The house width is \(120\) ft. Since the roof forms a \(45^{\circ}\) angle with the top of the house, we can consider two congruent right - triangles (dividing the roof along the vertical axis of symmetry of the house). The base of each right - triangle is \(b=\frac{120}{2} = 60\) ft.

Step2: Use trigonometry or properties of a \(45 - 45-90\) triangle

In a \(45 - 45-90\) triangle, the ratio of the sides is \(1:1:\sqrt{2}\). Let the length of the hypotenuse (roof edge of one side) be \(l\). If the leg (base of the right - triangle) \(a = 60\) ft, and for a \(45 - 45-90\) triangle \(l=a\sqrt{2}\).
The total length of the roof edge (sum of the two hypotenuses of the congruent right - triangles) \(L = 2l\). Substituting \(l = 60\sqrt{2}\) (from the property of \(45 - 45-90\) triangle where \(l\) (hypotenuse) and \(a\) (leg) satisfy \(l=a\sqrt{2}\) with \(a = 60\)), we get \(L=120\sqrt{2}\) ft.

Answer:

\(120\sqrt{2}\text{ ft}\)