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 a rectangle. Let's find the side - lengths.
If we assume the vertices of the stage (a rectangle) have coordinates \(A(8,60)\), \(F(16,60)\), \(C(64,8)\), \(E(72,24)\) (assuming the grid has a scale where each square is 4 units). Wait, no, let's re - check.
Assume each grid square is 4 units.
For the stage (a rectangle \(BCFE\)):
The length \(BC\): \(x\) - coordinate difference. \(B(72,60)\), \(C(64,8)\). No, wrong. Wait, the stage is a rectangle. Let's assume the vertices of the stage (rectangle) are \(B(72,60)\), \(C(64,8)\), \(E(72,24)\), \(F(16,60)\). No, better way:
Count the number of units on each side.
For the perimeter of the stage (rectangle):
Length \(l\): The horizontal side. If we count the number of 4 - unit segments. Let's assume from \(x = 16\) to \(x = 72\) (for the top side of the non - shaded rectangle). The number of 4 - unit segments: \(\frac{72 - 16}{4}=14\) units (length). The vertical side: from \(y = 8\) to \(y = 60\). The number of 4 - unit segments: \(\frac{60 - 8}{4}=13\) units.
Perimeter of a rectangle \(P = 2(l + w)\). Here \(l=(72 - 16)\) (in units of 4 - foot segments) and \(w=(60 - 8)\) (in units of 4 - foot segments).
\(P=2((72 - 16)+(60 - 8))\)
\(=2(56 + 52)\)
\(=2\times108=216\) feet.

Step2: Find the area of the dining section (trapezoid)

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 parallel sides: \(a=(60 - 8)=52\) (length of the vertical side of the trapezoid, in 4 - foot units) and \(b=(60 - 60)=0\) (no, wrong). Wait, the two parallel sides of the trapezoid:
Let's use the formula \(A=\frac{1}{2}(base_1+base_2)\times height\).
Base1: The length of \(AD\) (vertical side). \(AD=(60 - 8)=52\) (in 4 - foot units). Base2: The length of \(BC\) (vertical side). \(BC=(60 - 60) = 0\) (no). Wait, better:
Count the number of 4 - foot units.
The two parallel sides of the trapezoid:
One parallel side \(a=(72 - 16)=56\) (horizontal side, in 4 - foot units) and the other parallel side \(b=(64 - 16)=48\) (horizontal side, in 4 - foot units). The height \(h=(60 - 24)=36\) (vertical side, in 4 - foot units).
\(A=\frac{(56 + 48)\times36}{2}\)
\(=\frac{104\times36}{2}=104\times18 = 1872\) square feet.

Answer:

The perimeter of the stage is \(216\) feet.
The area of the dining section is \(1872\) square feet.