Question
Use the coordinates to compute the approximate area of the rectangle. Round your answer to the nearest whole number. The coordinates are,
D: (0, 14)
C: (6, 16)
B: (10, 4)
A: (4, 2)
D: (0, 14)
C: (6, 16)
B: (10, 4)
A: (4, 2)
Answers
Answer
The test says "140" is wrong.
Answered by
GPT-4o mini
To find the area of the rectangle formed by the given coordinates, we first need to determine the lengths of the sides.
Let's identify the coordinates:
- A: (4, 2)
- B: (10, 4)
- C: (6, 16)
- D: (0, 14)
Based on the arrangement of these points, we will calculate the lengths of the sides along the x-axis and y-axis.
1. Determine the x-coordinates of the leftmost and rightmost points (this will give us one length):
- The x-coordinates are: 0 (D), 4 (A), 6 (C), and 10 (B).
- The leftmost point is 0 (D), and the rightmost point is 10 (B).
- Length on the x-axis = rightmost x - leftmost x = 10 - 0 = 10.
2. Determine the y-coordinates of the bottommost and topmost points (this will give us the other length):
- The y-coordinates are: 2 (A), 4 (B), 14 (D), and 16 (C).
- The bottommost point is 2 (A), and the topmost point is 16 (C).
- Length on the y-axis = topmost y - bottommost y = 16 - 2 = 14.
3. Now, we can compute the area of the rectangle:
\[
\text{Area} = \text{length} \times \text{width} = 10 \times 14 = 140.
\]
Thus, the approximate area of the rectangle is \( \boxed{140} \).
Let's identify the coordinates:
- A: (4, 2)
- B: (10, 4)
- C: (6, 16)
- D: (0, 14)
Based on the arrangement of these points, we will calculate the lengths of the sides along the x-axis and y-axis.
1. Determine the x-coordinates of the leftmost and rightmost points (this will give us one length):
- The x-coordinates are: 0 (D), 4 (A), 6 (C), and 10 (B).
- The leftmost point is 0 (D), and the rightmost point is 10 (B).
- Length on the x-axis = rightmost x - leftmost x = 10 - 0 = 10.
2. Determine the y-coordinates of the bottommost and topmost points (this will give us the other length):
- The y-coordinates are: 2 (A), 4 (B), 14 (D), and 16 (C).
- The bottommost point is 2 (A), and the topmost point is 16 (C).
- Length on the y-axis = topmost y - bottommost y = 16 - 2 = 14.
3. Now, we can compute the area of the rectangle:
\[
\text{Area} = \text{length} \times \text{width} = 10 \times 14 = 140.
\]
Thus, the approximate area of the rectangle is \( \boxed{140} \).
Answered by
GPT-4o mini
I apologize for the confusion. Let's re-evaluate the coordinates and clarify how they define a rectangle or a polygon.
The coordinates are:
- D: (0, 14)
- C: (6, 16)
- B: (10, 4)
- A: (4, 2)
### Step 1: Identify the shape
On further inspection of the coordinates, it appears that these points do not form a rectangle directly. Let's analyze the coordinates and find the area by breaking it down into triangles or using the shoelace formula instead.
### Step 2: Use the Shoelace Formula
The Shoelace theorem gives a straightforward way to calculate the area of a simple polygon when the vertices are known.
The formula for the area is:
\[
\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1}) \right|
\]
where \((x_{n+1}, y_{n+1})\) is the same as \((x_1, y_1)\).
### Step 3: List the points in order
We will list the points in a specific order, either clockwise or counterclockwise. Let's arrange them as follows based on their positions:
1. D: (0, 14)
2. C: (6, 16)
3. B: (10, 4)
4. A: (4, 2)
5. (0, 14) again to close the polygon.
Now, we will calculate the area using the Shoelace formula.
### Step 4: Applying the formula
\[
\text{Area} = \frac{1}{2} \left| (0 \cdot 16 + 6 \cdot 4 + 10 \cdot 2 + 4 \cdot 14) - (14 \cdot 6 + 16 \cdot 10 + 4 \cdot 4 + 2 \cdot 0) \right|
\]
Calculating each term:
- First sum:
\[
0 \cdot 16 + 6 \cdot 4 + 10 \cdot 2 + 4 \cdot 14 = 0 + 24 + 20 + 56 = 100
\]
- Second sum:
\[
14 \cdot 6 + 16 \cdot 10 + 4 \cdot 4 + 2 \cdot 0 = 84 + 160 + 16 + 0 = 260
\]
Now substituting these sums back into the formula:
\[
\text{Area} = \frac{1}{2} \left| 100 - 260 \right| = \frac{1}{2} \left| -160 \right| = \frac{1}{2} \times 160 = 80
\]
### Final Result
Thus, the area of the polygon formed by the points A, B, C, and D is \( \boxed{80} \). Thank you for your patience, and I hope this resolves the issue!
The coordinates are:
- D: (0, 14)
- C: (6, 16)
- B: (10, 4)
- A: (4, 2)
### Step 1: Identify the shape
On further inspection of the coordinates, it appears that these points do not form a rectangle directly. Let's analyze the coordinates and find the area by breaking it down into triangles or using the shoelace formula instead.
### Step 2: Use the Shoelace Formula
The Shoelace theorem gives a straightforward way to calculate the area of a simple polygon when the vertices are known.
The formula for the area is:
\[
\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1}) \right|
\]
where \((x_{n+1}, y_{n+1})\) is the same as \((x_1, y_1)\).
### Step 3: List the points in order
We will list the points in a specific order, either clockwise or counterclockwise. Let's arrange them as follows based on their positions:
1. D: (0, 14)
2. C: (6, 16)
3. B: (10, 4)
4. A: (4, 2)
5. (0, 14) again to close the polygon.
Now, we will calculate the area using the Shoelace formula.
### Step 4: Applying the formula
\[
\text{Area} = \frac{1}{2} \left| (0 \cdot 16 + 6 \cdot 4 + 10 \cdot 2 + 4 \cdot 14) - (14 \cdot 6 + 16 \cdot 10 + 4 \cdot 4 + 2 \cdot 0) \right|
\]
Calculating each term:
- First sum:
\[
0 \cdot 16 + 6 \cdot 4 + 10 \cdot 2 + 4 \cdot 14 = 0 + 24 + 20 + 56 = 100
\]
- Second sum:
\[
14 \cdot 6 + 16 \cdot 10 + 4 \cdot 4 + 2 \cdot 0 = 84 + 160 + 16 + 0 = 260
\]
Now substituting these sums back into the formula:
\[
\text{Area} = \frac{1}{2} \left| 100 - 260 \right| = \frac{1}{2} \left| -160 \right| = \frac{1}{2} \times 160 = 80
\]
### Final Result
Thus, the area of the polygon formed by the points A, B, C, and D is \( \boxed{80} \). Thank you for your patience, and I hope this resolves the issue!