Question
Plot and Find Area of Polygon (Rectilinear Lengths)
Score: 0/3
Penalty: none
Question
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Plot and connect the points in the order listed below. When you are done, find the area of the resulting figure.
A, left parenthesis, minus, 3, comma, 2, right parenthesisA(−3,2), B, left parenthesis, 5, comma, 2, right parenthesisB(5,2), C, left parenthesis, 5, comma, minus, 7, right parenthesisC(5,−7), D, left parenthesis, minus, 3, comma, minus, 7, right parenthesisD(−3,−7), E, left parenthesis, minus, 3, comma, minus, 4, right parenthesisE(−3,−4), F, of, 1, comma, minus, 4F(1,−4), G, of, minus, 3, comma, minus, 1G(−3,−1)
Score: 0/3
Penalty: none
Question
Show Examples
Plot and connect the points in the order listed below. When you are done, find the area of the resulting figure.
A, left parenthesis, minus, 3, comma, 2, right parenthesisA(−3,2), B, left parenthesis, 5, comma, 2, right parenthesisB(5,2), C, left parenthesis, 5, comma, minus, 7, right parenthesisC(5,−7), D, left parenthesis, minus, 3, comma, minus, 7, right parenthesisD(−3,−7), E, left parenthesis, minus, 3, comma, minus, 4, right parenthesisE(−3,−4), F, of, 1, comma, minus, 4F(1,−4), G, of, minus, 3, comma, minus, 1G(−3,−1)
Answers
GPT-4o mini
To plot the points and find the area of the polygon defined by the points A, B, C, D, E, F, and G in the order you specified, we will follow these steps:
### 1. Plotting the Points
The points provided are:
- A(-3, 2)
- B(5, 2)
- C(5, -7)
- D(-3, -7)
- E(-3, -4)
- F(1, -4)
- G(-3, -1)
Now we will connect the points in the given order:
1. Start at A (-3, 2)
2. Move to B (5, 2)
3. Down to C (5, -7)
4. Left to D (-3, -7)
5. Up to E (-3, -4)
6. Right to F (1, -4)
7. Up to G (-3, -1)
8. Finally back to A (-3, 2)
### 2. Visualizing the Polygon
The resulting figure will likely have the following shape:
- A rectangle from (-3, 2) to (5, -7), with vertical lines connecting A, B, C, and D.
- A small triangle or additional shape created by the points E, F, and G.
### 3. Calculating the Area
#### Using the Shoelace Formula
To find the area of the polygon defined by the vertices (in order), we can apply the Shoelace Theorem. The area \( A \) can be calculated as follows:
Given vertices:
- A(-3, 2)
- B(5, 2)
- C(5, -7)
- D(-3, -7)
- E(-3, -4)
- F(1, -4)
- G(-3, -1)
We can write the coordinates in the table form for the Shoelace formula:
\[
\begin{array}{c|c}
x & y \\
\hline
-3 & 2 \\
5 & 2 \\
5 & -7 \\
-3 & -7 \\
-3 & -4 \\
1 & -4 \\
-3 & -1 \\
\end{array}
\]
Now applying the Shoelace Formula:
\[
A = \frac{1}{2} \left| \sum (x_iy_{i+1}) - \sum (y_ix_{i+1}) \right|
\]
Calculating using our coordinates. We will loop back to the first vertex to close the shape:
1. Calculate \( \sum (x_iy_{i+1}) \):
- (-3)(2) + (5)(-7) + (5)(-7) + (-3)(-4) + (-3)(-4) + (1)(-1) + (-3)(2)
- = -6 -35 - 35 + 12 + 12 - 1 - 6
- = -69
2. Calculate \( \sum (y_ix_{i+1}) \):
- (2)(5) + (2)(5) + (-7)(-3) + (-7)(-3) + (-4)(1) + (-4)(-3) + (-1)(-3)
- = 10 + 10 + 21 + 21 - 4 + 12 + 3
- = 73
Now substituting back into the area formula:
\[
A = \frac{1}{2} \left| -69 - 73 \right| = \frac{1}{2} \left| -142 \right| = \frac{1}{2} \times 142 = 71
\]
### Final Result
Thus, the area of the polygon formed by the points A(-3, 2), B(5, 2), C(5, -7), D(-3, -7), E(-3, -4), F(1, -4), G(-3, -1) is **71 square units**.
### 1. Plotting the Points
The points provided are:
- A(-3, 2)
- B(5, 2)
- C(5, -7)
- D(-3, -7)
- E(-3, -4)
- F(1, -4)
- G(-3, -1)
Now we will connect the points in the given order:
1. Start at A (-3, 2)
2. Move to B (5, 2)
3. Down to C (5, -7)
4. Left to D (-3, -7)
5. Up to E (-3, -4)
6. Right to F (1, -4)
7. Up to G (-3, -1)
8. Finally back to A (-3, 2)
### 2. Visualizing the Polygon
The resulting figure will likely have the following shape:
- A rectangle from (-3, 2) to (5, -7), with vertical lines connecting A, B, C, and D.
- A small triangle or additional shape created by the points E, F, and G.
### 3. Calculating the Area
#### Using the Shoelace Formula
To find the area of the polygon defined by the vertices (in order), we can apply the Shoelace Theorem. The area \( A \) can be calculated as follows:
Given vertices:
- A(-3, 2)
- B(5, 2)
- C(5, -7)
- D(-3, -7)
- E(-3, -4)
- F(1, -4)
- G(-3, -1)
We can write the coordinates in the table form for the Shoelace formula:
\[
\begin{array}{c|c}
x & y \\
\hline
-3 & 2 \\
5 & 2 \\
5 & -7 \\
-3 & -7 \\
-3 & -4 \\
1 & -4 \\
-3 & -1 \\
\end{array}
\]
Now applying the Shoelace Formula:
\[
A = \frac{1}{2} \left| \sum (x_iy_{i+1}) - \sum (y_ix_{i+1}) \right|
\]
Calculating using our coordinates. We will loop back to the first vertex to close the shape:
1. Calculate \( \sum (x_iy_{i+1}) \):
- (-3)(2) + (5)(-7) + (5)(-7) + (-3)(-4) + (-3)(-4) + (1)(-1) + (-3)(2)
- = -6 -35 - 35 + 12 + 12 - 1 - 6
- = -69
2. Calculate \( \sum (y_ix_{i+1}) \):
- (2)(5) + (2)(5) + (-7)(-3) + (-7)(-3) + (-4)(1) + (-4)(-3) + (-1)(-3)
- = 10 + 10 + 21 + 21 - 4 + 12 + 3
- = 73
Now substituting back into the area formula:
\[
A = \frac{1}{2} \left| -69 - 73 \right| = \frac{1}{2} \left| -142 \right| = \frac{1}{2} \times 142 = 71
\]
### Final Result
Thus, the area of the polygon formed by the points A(-3, 2), B(5, 2), C(5, -7), D(-3, -7), E(-3, -4), F(1, -4), G(-3, -1) is **71 square units**.