Question

select the correct answer from each drop - down menu. a banquet hall is mapped on the grid below, with lengths measured in feet. one corner of the banquet hall is used for the stage. the orange shaded region represents the dining section. complete the sentences describing the banquet hall. the perimeter of the stage is feet. the area of the dining section is square feet

Explanation:

Step1: Find the coordinates of the stage

Assume the stage is the rectangle \(FBC E\). Let \(F=(16,60)\), \(B=(72,60)\), \(C=(72,8)\), \(E=(64,24)\).
The length \(FB = 72 - 16=56\) feet, the length \(BC=60 - 8 = 52\) feet.
The perimeter of a rectangle \(P = 2(l + w)\). For rectangle \(FBCE\), \(l = 56\), \(w = 52\).
\(P=2(56 + 52)=2\times108 = 216\) feet.

Step2: Find the area of the dining - section (trapezoid \(AD EF\))

Let \(A=(8,60)\), \(D=(8,8)\), \(E=(64,24)\), \(F=(16,60)\).
The formula for the area of a trapezoid is \(A=\frac{(a + b)h}{2}\), where \(a\) and \(b\) are the lengths of the parallel sides and \(h\) is the height.
The length of \(AD=60 - 8=52\) feet, the length of \(EF\):
Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) for \(E=(64,24)\) and \(F=(16,60)\)
\(EF=\sqrt{(64 - 16)^2+(24 - 60)^2}=\sqrt{48^2+(- 36)^2}=\sqrt{(12\times4)^2+(12\times(-3))^2}=12\sqrt{16 + 9}=12\times5 = 60\) feet.
The horizontal distance between \(A=(8,60)\) and \(F=(16,60)\) (which is the height of the trapezoid) \(h = 16 - 8=8\) units. But if we consider the trapezoid \(ADEF\) with parallel sides \(AD = 52\) and \(EF = 60\) and the base - length (horizontal distance from \(D=(8,8)\) to \(E=(64,24)\) in the direction perpendicular to the parallel sides).
Another way: Counting the grid - based on the coordinates.
The formula for the area of a trapezoid \(A=\frac{(y_A - y_D+y_E - y_C)}{2}\times(x_E - x_D)\)
\(y_A - y_D=60 - 8 = 52\), \(y_E - y_C=24 - 8=16\), \(x_E - x_D=64 - 8 = 56\)
\(A=\frac{(52 + 16)}{2}\times56=\frac{68}{2}\times56=34\times56 = 1904\) square feet.

Answer:

The perimeter of the stage is \(216\) feet.
The area of the dining section is \(1904\) square feet.