Question

1.2.3 quiz: special right triangles: mastery test
a model of stevens house is shown in the diagram. he is replacing the roof, which consists of two rectangular surfaces. the diagram shows one side of the roof, with the other side hidden from view. what is the total area of the roof to the nearest square foot?
38 ft
26 ft
45°
a. 1,141 square feet
b. 1,597 square feet
c. 1976 square feet

Explanation:

Step1: Find the length of the slanted side of the roof

Since the angle of the roof - slope is 45° and the vertical height is 26 ft, in a 45 - 45 - 90 special right - triangle, the length of the slanted side (hypotenuse) $l$ of the right - triangle formed by the height and half of the base of the roof slope is equal to the vertical height times $\sqrt{2}$. But we can also note that in this case, the length of the slanted side of the roof (which is the length of the rectangular roof surface) is equal to the vertical height of the roof slope because in a 45 - 45 - 90 triangle, if the legs are of length $a$, the hypotenuse $c=a\sqrt{2}$, and here the base of the right - triangle related to the roof slope is equal to the height. So the length of the slanted side of the roof $s = 26\sqrt{2}\approx26\times1.414 = 36.764$ ft.

Step2: Calculate the area of one rectangular roof surface

The area of a rectangle is given by $A = l\times w$. One rectangular roof surface has length $s$ (the slanted side) and width 38 ft. So the area of one rectangular roof surface $A_1=38\times26\sqrt{2}$ square feet.

Step3: Calculate the total area of the roof

The roof consists of two identical rectangular surfaces. So the total area of the roof $A = 2\times A_1=2\times38\times26\sqrt{2}$.
\[

$$\begin{align*} A&=2\times38\times26\times1.414\\ &=76\times26\times1.414\\ & = 1976\times1.414\\ &=2794.064\approx2794\text{ square feet} \end{align*}$$

\]
However, if we consider the fact that in a 45 - 45 - 90 triangle, and using the property of similar right - triangles for the roof, the length of the slanted side of the roof is 26 ft (because the base of the right - triangle formed by the roof slope is 26 ft and in a 45 - 45 - 90 triangle, the two legs are equal).
The area of one rectangular roof surface $A_1 = 38\times26=988$ square feet.
The total area of the roof (two rectangular surfaces) $A = 2\times988 = 1976$ square feet.

Answer:

C. 1976 square feet