A wall in Maria's bedroom is in the shape of a trapezoid. The wall can be divided into a rectangle and a triangle.
Using the $45^{\circ}-45^{\circ}-90^{\circ}$ triangle theorem, find the value of $h$, the height of the wall.
A. 6.5 ft
B. $6.5 \sqrt{2} ft$
C. 13 ft
D. $13 \sqrt{2} ft$
Answer (1)
Analyze the properties of a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, where the legs are equal and the hypotenuse is 2 times the leg length.
Consider each option for the height h and calculate the corresponding hypotenuse.
Recognize that without more information, we cannot definitively determine the height, but we can relate it to the hypotenuse.
Based on the options provided, the most straightforward answer is: 6.5 f t
Explanation
Problem Analysis We are given a trapezoidal wall that can be divided into a rectangle and a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle. We need to find the height, h , of the wall using the properties of the special right triangle. In a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, the two legs are equal, and the hypotenuse is 2 times the length of each leg.
Analyzing the Options Let's analyze each of the given options for the height h :
If h = 6.5 ft, then the other leg of the triangle is also 6.5 ft, and the hypotenuse is 6.5 2 ft ≈ 9.19 ft.
If h = 6.5 2 ft, then the other leg is also 6.5 2 ft, and the hypotenuse is 6.5 2 ⋅ 2 = 6.5 ⋅ 2 = 13 ft.
If h = 13 ft, then the other leg is also 13 ft, and the hypotenuse is 13 2 ft ≈ 18.38 ft.
If h = 13 2 ft, then the other leg is also 13 2 ft, and the hypotenuse is 13 2 ⋅ 2 = 13 ⋅ 2 = 26 ft.
Relating Height and Hypotenuse Without additional information about the dimensions of the trapezoid or the triangle (such as the length of the hypotenuse), we cannot uniquely determine the height h . However, we can express the relationship between the height and the hypotenuse. If we assume that the length of one of the legs of the 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle is equal to the height of the wall, then the hypotenuse is 2 times the height.
Considering Possible Values Since we don't have enough information to determine the exact value of h , we must rely on the given options. If the base of the triangle is 6.5 ft, then the height is 6.5 ft and the hypotenuse is 6.5 2 ft. If the height is 6.5 2 ft, then the base is 6.5 2 ft and the hypotenuse is 13 ft.
Final Considerations The problem states that we need to find the value of h , the height of the wall. Since the triangle is a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, the height and the base are equal. If the height is 6.5 ft, the hypotenuse is 6.5 2 ft. If the height is 6.5 2 ft, the hypotenuse is 13 ft. Without more information, we cannot definitively choose one value over the other. However, if we assume the shorter leg is 6.5, then the height is 6.5. If we assume the longer leg (hypotenuse) is 13, then the height is 6.5 2 .
Conclusion Without more information, it's impossible to determine the exact height. However, based on the given options, if one leg of the 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle is 6.5 ft, then the height of the wall is 6.5 ft and the hypotenuse is 6.5 2 ft. If the hypotenuse is 13 ft, then the height is 6.5 2 ft. Since the options include both 6.5 ft and 6.5 2 ft, we cannot definitively choose one. However, if the question implies that the given options are the only possible values for the height, then we can assume that the height is either 6.5 ft or 6.5 2 ft.
Final Answer Since we don't have enough information to determine the exact value of h, we must rely on the given options. If the base of the triangle is 6.5 ft, then the height is 6.5 ft and the hypotenuse is 6.5 2 ft. If the height is 6.5 2 ft, then the base is 6.5 2 ft and the hypotenuse is 13 ft. Without additional information, we cannot definitively choose one value over the other. However, if we assume the shorter leg is 6.5, then the height is 6.5. If we assume the longer leg (hypotenuse) is 13, then the height is 6.5 2 .
Based on the options provided, the most straightforward answer is:
6.5 f t
Examples
Understanding 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangles is useful in construction and design. For example, when building a ramp with a 4 5 ∘ angle, knowing the height allows you to easily calculate the length of the ramp (hypotenuse) using the relationship h y p o t e n u se = h e i g h t × 2 . This ensures accurate and safe construction.
