A hip roof is shown in the diagram. The roof front and back are in the shape of congruent isosceles trapezoids. The two sides are in the shape of congruent isosceles triangles. Determine the total roof area.
The roof front and back are congruent isosceles trapezoids, which means they have parallel bases with equal lengths. Since they are congruent, the total area of the front and back of the roof will be twice the area of one of them.
The two sides are congruent isosceles triangles. Since they are congruent, the total area of both sides will be twice the area of one of them.
To find the total roof area, we need to find the areas of the front and back, as well as the areas of both sides, and then add them up.
Let's determine the area of the trapezoid first. We can use the formula for the area of a trapezoid, which is $\frac{1}{2}(b_1 + b_2)h$, where $b_1$ and $b_2$ are the lengths of the bases, and $h$ is the height.
Since the roof front and back are congruent trapezoids, their bases and heights are equal.
Let's label the bases of the trapezoid as $b_1$ and $b_2$, and let's label the height as $h$.
The total area of the front and back will be $2\left(\frac{1}{2}(b_1 + b_2)h\right) = (b_1 + b_2)h$.
Now let's determine the area of the triangles. The area of an isosceles triangle can be found using the formula $\frac{1}{2}bh$, where $b$ is the length of the base and $h$ is the height.
Since the sides of the roof are congruent triangles, their bases and heights are equal.
Let's label the base of one of the triangles as $b$ and the height as $h$.
The total area of both sides will be $2 \left(\frac{1}{2} bh\right) = bh$.
To find the total roof area, we need to add up the area of the front, back, and both sides.
Total roof area = (area of front and back) + (area of both sides) = $(b_1 + b_2)h + bh = h(b_1 + b_2 + b)$.
Since the diagram and problem statement do not provide any numerical values for the bases or height, we cannot determine the exact total roof area. We can only express it as the sum of the three lengths $b_1$, $b_2$, and $b$, multiplied by the height $h$.