The figure below represents a large poster being designed for a pep rally in the school gym. The units on the grid represent meters.

Question

On the coordinate plane, the point A is at (4, 2), B is at (4, 12), C is at (8, 12), D is at (8, 2), E is at (2, 2), F is at (6, 18), and G is at (10, 2).

What is the total area of the poster?

Use the drop-down menus to choose the correct response that makes each statement true.

The bottom portion of the figure, quadrilateral , may be composed into a larger rectangle with base

. The area of this composed rectangle is Query 1
square meters.
The triangles that need to be subtracted from the composed rectangle to find the area bounded by each have a base of Query 2
meters and a height of Query 3
meters.
The area of triangle is Query 4
square meters.
The total area of the figure is Query 5
square meters.

Answer 1

To find the total area of the poster designed for the pep rally, we must break it down into parts according to the coordinate points provided.

Step 1: Identify the shapes and their dimensions

1. Quadrilateral ABCD:

   • Points: A(4, 2), B(4, 12), C(8, 12), D(8, 2)
   • This shape is a rectangle with:
     • Base = $C_x-A_x=8-4=4$ meters
     • Height = $B_y-A_y=12-2=10$ meters
   • Area of rectangle ABCD: $${\text{Area}}_{ABCD}=\text{Base}\times \text{Height}=4,\text{m}\times 10,\text{m}=40,\text{square meters}$$

2. Triangle EFG:

   • Points: E(2, 2), F(6, 18), G(10, 2)
   • This triangle has:
     • Base = $G_x-E_x=10-2=8$ meters (the line segment between E and G)
     • Height = $F_y-E_y=18-2=16$ meters (the vertical distance from point F down to the line containing E and G)
   • Area of triangle EFG: $${\text{Area}}_{EFG}=\frac{1}{2}\times \text{Base}\times \text{Height}=\frac{1}{2}\times 8,\text{m}\times 16,\text{m}=64,\text{square meters}$$

Step 2: Calculate total area

Now, since the total area of the poster includes the area of rectangle ABCD and the area of triangle EFG, we can combine these areas:

\[ \text{Total Area} = \text{Area}{ABCD} + \text{Area}{EFG} = 40 , \text{square meters} + 64 , \text{square meters} = 104 , \text{square meters} \]

Summary for the Queries

• Query 1: The area of the composed rectangle (the area of quadrilateral ABCD) is 40 square meters.
• Query 2: The base of the triangles that need to be subtracted is 8 meters.
• Query 3: The height of the triangles is 16 meters.
• Query 4: The area of triangle EFG is 64 square meters.
• Query 5: The total area of the figure is 104 square meters.

Final Responses

1. The bottom portion of the figure, quadrilateral ABCD, may be composed into a larger rectangle with base 4. The area of this composed rectangle is 40 square meters.
2. The triangles that need to be subtracted from the composed rectangle to find the area bounded by each have a base of 8 meters and a height of 16 meters.
3. The area of triangle EFG is 64 square meters.
4. The total area of the figure is 104 square meters.

Answer 2

1. The bottom portion of the figure, quadrilateral BCGE, may be composed into a larger rectangle with base **EG**. The area of this composed rectangle is **40** square meters.
2. The triangles that need to be subtracted from the composed rectangle to find the area bounded by BCGE each have a base of **8** meters and a height of **16** meters.
3. The area of triangle BFC is **64** square meters.
4. The total area of the figure is **104** square meters.

Answer 3

Let's clarify the details based on your responses regarding the poster represented in the coordinate plane:

Identifying the bottom portion

1. The bottom portion of the figure, quadrilateral BCGE, may be composed into a larger rectangle with base EG. The correct identification of the rectangle is important.
   • Base EG: From point E(2, 2) to point G(10, 2), the distance (or base) is given by $G_x-E_x=10-2=8$ meters.
   • Height of BCGE:
     • The height of the rectangle ABCD (which extends from B(4, 12) to E(2, 2)) is the vertical line segment from the line $y=2$ to the line $y=12$ (or the y-coordinate of point B), which is $12-2=10$ meters.
   • So, the area of rectangle BCGE: $${\text{Area}}_{BCGE}=\text{Base}\times \text{Height}=8,\text{m}\times 10,\text{m}=80,\text{square meters}$$

Identifying triangles for calculation

2. The triangles that need to be subtracted from the composed rectangle to find the area bounded by BCGE
   • It seems there was a misunderstanding in your statements. The triangles to subtract should be specifically referencing triangles that form the upper parts of the rectangle. This likely refers to triangles like BFC (above base EG).
   • Each of these triangles would still have a base corresponding to their respective dimensions based on the coordinates you provide specifically for BCGE.

Area calculations

3. Area of triangle BFC: From point B(4, 12) to C(8, 12) to F(6, 18), where the height to the base BC (at y=12) would indeed yield: $$\text{Base}=BC=|C_x-B_x|=8-4=4,\text{m}$$ and the height is: $$\text{Height}=F_y-B_y=18-12=6,\text{m}$$ So, the area of triangle BFC can be computed as: $${\text{Area}}_{BFC}=\frac{1}{2}\times 4\times 6=12,\text{square meters}$$

Total area recalculation

4. The total area calculation should sum the areas of the respective components. If we consider the rectangle BCGE and the triangle (subtracting triangles properly considered), \[ \text{Total Area} = \text{Area}{BCGE} + \text{Area}{BFC} = 80 + 12 = 92 , \text{square meters} \]

Final Responses Update

1. The bottom portion of the figure, quadrilateral BCGE, may be composed into a larger rectangle with base EG. The area of this composed rectangle is 80 square meters.
2. The triangles that need to be subtracted from the composed rectangle to find the area bounded by BCGE each have a base of 4 meters and a height of 6 meters.
3. The area of triangle BFC is 12 square meters (not 64 as stated earlier).
4. The total area of the figure, after considering all relevant components, is 92 square meters.

Please adjust the associated areas based on corrections needed in the descriptions to align correctly with the figures represented!

Answer 4

meters and a height of **6** meters.
3. The area of triangle BFC is **12** square meters (not **64** as stated earlier).
4. The total area of the figure, after considering all relevant components, is **92** square meters.

all dat up there is wrong |^

Answer 5

I apologize for any confusion in my previous messages. Let's go through the geometry of the problem step by step to clarify the areas once more based on the coordinates you've provided and arrive at the correct calculations.

Step 1: Identify Points and Shapes Accurately

The points you provided are:

• A(4, 2), B(4, 12), C(8, 12), D(8, 2), E(2, 2), F(6, 18), and G(10, 2).

Step 2: Analyzing the Shapes

1. Quadrilateral BCGE:
   • Points for BCGE: B(4, 12), C(8, 12), G(10, 2), E(2, 2)
   • The area needs to be computed as follows:
   • The shape BCGE forms at the bottom of points B, C, E, and G.

Step 3: Area of Rectangle BCGE

• Base (EG):

  • The distance from E(2, 2) to G(10, 2) is: $$\text{Base}=G_x-E_x=10-2=8,\text{meters}$$

• Height is the vertical distance from the bottom (y = 2) to the top (y = 12):

  • The height from the line at y=2 to the line at y=12 is: $$\text{Height}=B_y-E_y=12-2=10,\text{meters}$$

• So, the area of rectangle BCGE: $${\text{Area}}_{BCGE}=\text{Base}\times \text{Height}=8,\text{m}\times 10,\text{m}=80,\text{square meters}$$

Step 4: Area of Triangle BFC

• Triangle BFC has points B(4, 12), F(6, 18), and C(8, 12):

  • Base (BC) spans horizontally from B(4, 12) to C(8, 12): $$\text{Base}=|C_x-B_x|=|8-4|=4,\text{meters}$$

• The height is:

  • From point F(6, 18) down vertically to line BC (y = 12): $$\text{Height}=F_y-B_y=18-12=6,\text{meters}$$

So, the area of triangle BFC: $${\text{Area}}_{BFC}=\frac{1}{2}\times \text{Base}\times \text{Height}=\frac{1}{2}\times 4,\text{m}\times 6,\text{m}=12,\text{square meters}$$

Step 5: Total Area of the Figure

• Finally, the total area of the figure can be calculated as combines: \[ \text{Total Area} = \text{Area}{BCGE} + \text{Area}{BFC} = 80 , \text{square meters} + 12 , \text{square meters} = 92 , \text{square meters} \]

Summary Statements

1. The bottom portion of the figure, quadrilateral BCGE, may be composed into a larger rectangle with base 8 meters. The area of this composed rectangle is 80 square meters.
2. The triangles that need to be subtracted from the composed rectangle to find the area bounded by BCGE each have a base of 4 meters and a height of 6 meters.
3. The area of triangle BFC is 12 square meters.
4. The total area of the figure is 92 square meters.

If you have sentences or areas that need adjustments that I may not have confused or clarified properly, please specify those and let's ensure we're aligned correctly! Thank you for your patience.