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.
Answer (1)
Recognize the wall includes a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, where the height h is one leg.
Use the 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle theorem to relate the leg and hypotenuse: h y p o t e n u se = h 2 .
Test the options to find a consistent leg-hypotenuse pair within the given choices.
Determine that if the hypotenuse is 13 ft, then h = 2 13 = 6.5 2 ft, which is a consistent solution: 6.5 2 f t .
Explanation
Problem Analysis The problem states that a wall is in the shape of a trapezoid, which can be divided into a rectangle and a triangle. We are asked to find the height of the wall, h , using the 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle theorem. This means the triangle formed is a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle.
4 5 ∘ − 4 5 ∘ − 9 0 ∘ Triangle Theorem In a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, the two legs are equal in length, and the hypotenuse is 2 times the length of each leg. Let x be the length of each leg of the 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle. Then, the hypotenuse is x 2 . The height of the wall, h , corresponds to the length of a leg of the 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, so h = x . We need to determine which of the given options represents the length of the hypotenuse, allowing us to solve for h .
Analyzing the Options Let's analyze each option:
If h = 6.5 ft, then the hypotenuse would be 6.5 2 ft. This matches one of the options.
If h = 6.5 2 ft, then the hypotenuse would be ( 6.5 2 ) 2 = 6.5 ∗ 2 = 13 ft. This also matches one of the options.
If h = 13 ft, then the hypotenuse would be 13 2 ft. This matches another option.
If h = 13 2 ft, then the hypotenuse would be ( 13 2 ) 2 = 13 ∗ 2 = 26 ft. This does not match any of the options.
Finding the Correct Height Since the problem provides four possible values for the height h , and we are given that the triangle is a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, we can test each option to see if the corresponding hypotenuse is also among the options. If h = 6.5 , the hypotenuse is 6.5 2 . If h = 6.5 2 , the hypotenuse is 13 . If h = 13 , the hypotenuse is 13 2 . If h = 13 2 , the hypotenuse is 26 , which is not in the options. Therefore, we can consider the first three cases as valid possibilities for the height h . However, without more information, we cannot determine a unique value for h .
However, if we assume that the problem is well-posed and has a unique solution among the given options, we must look for a case where both the leg and the hypotenuse are among the options. This occurs when h = 6.5 2 , which gives a hypotenuse of 13 .
Calculating the Height If the hypotenuse of the triangle is 13 ft, then the height h (which is the length of a leg) can be found using the relationship h 2 = 13 . Solving for h , we get: h = 2 13 = 2 13 2 = 6.5 2 Thus, the height of the wall is 6.5 2 ft.
Final Answer Therefore, the value of h , the height of the wall, is 6.5 2 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 relationship between the legs and hypotenuse allows you to calculate the necessary materials. If you want the ramp to have a height of 6.5 2 feet, you know the ramp's length (hypotenuse) will be 13 feet, ensuring the ramp meets the required specifications.
