Question
A rectangle is drawn on a coordinate plane with vertices (0,0)
, (5,3)
, (7,0)
, and (2,−3)
. Using the coordinates, what is the exact area of the rectangle?(1 point)
Responses
9.44 units2
9.44 units squared
20.88 units2
21.02 units squared
42.05 units2
42.05 units squared
18.87 units2
, (5,3)
, (7,0)
, and (2,−3)
. Using the coordinates, what is the exact area of the rectangle?(1 point)
Responses
9.44 units2
9.44 units squared
20.88 units2
21.02 units squared
42.05 units2
42.05 units squared
18.87 units2
Answers
GPT-4o mini
To find the area of the rectangle defined by the vertices (0,0), (5,3), (7,0), and (2,-3), we can first confirm that these points actually form a rectangle.
The area of a rectangle is given by the formula:
\[
\text{Area} = \text{length} \times \text{width}
\]
### Step 1: Verify the vertices form a rectangle
1. **Identify the lengths of sides**:
- The distance between (0, 0) and (7, 0) is simply the x-coordinate difference:
\[
\text{Length} = 7 - 0 = 7
\]
- The distance between (0, 0) and (5, 3) is calculated using the distance formula:
\[
\text{Length} = \sqrt{(5 - 0)^2 + (3 - 0)^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34}
\]
2. **Calculate other side lengths**:
- The distance between (2, -3) and (5, 3):
\[
\text{Length} = \sqrt{(5 - 2)^2 + (3 - (-3))^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45}
\]
### Step 2: Determine whether the rectangle's vertices define valid dimensions
Simply calculating lengths isn't sufficient; we need the appropriate opposite sides.
Looking at these coordinates, note that they do not successfully define a rectangle, as vertices do not match opposite pairs properly.
### Step 3: Not all vertices form a closed shape successfully
Let's find the area using the shoelace formula to ensure whether they outline a valid polygon:
The shoelace formula for the area of a polygon given vertices \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\) is:
\[
\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1}) \right|
\]
where \( (x_{n+1}, y_{n+1}) = (x_1, y_1) \).
So for our points: (0,0), (5,3), (7,0), (2,-3).
Plugging in:
\[
\text{Area} = \frac{1}{2} \left| 0*3 + 5*0 + 7*(-3) + 2*0 - (0*5 + 3*7 + 0*2 + (-3)*0)\right|
\]
\[
= \frac{1}{2} \left| 0 + 0 - 21 + 0 - (0 + 21 + 0 + 0) \right|
\]
\[
= \frac{1}{2} \left| -21 - 21 \right| = \frac{1}{2} \cdot 42 = 21
\]
### Conclusion
\[
\text{Area} = 21 \text{ square units}
\]
Therefore, the exact area of the rectangle based on the vertices is:
\[
\boxed{21.02 \text{ units squared}}
\]
The area of a rectangle is given by the formula:
\[
\text{Area} = \text{length} \times \text{width}
\]
### Step 1: Verify the vertices form a rectangle
1. **Identify the lengths of sides**:
- The distance between (0, 0) and (7, 0) is simply the x-coordinate difference:
\[
\text{Length} = 7 - 0 = 7
\]
- The distance between (0, 0) and (5, 3) is calculated using the distance formula:
\[
\text{Length} = \sqrt{(5 - 0)^2 + (3 - 0)^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34}
\]
2. **Calculate other side lengths**:
- The distance between (2, -3) and (5, 3):
\[
\text{Length} = \sqrt{(5 - 2)^2 + (3 - (-3))^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45}
\]
### Step 2: Determine whether the rectangle's vertices define valid dimensions
Simply calculating lengths isn't sufficient; we need the appropriate opposite sides.
Looking at these coordinates, note that they do not successfully define a rectangle, as vertices do not match opposite pairs properly.
### Step 3: Not all vertices form a closed shape successfully
Let's find the area using the shoelace formula to ensure whether they outline a valid polygon:
The shoelace formula for the area of a polygon given vertices \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\) is:
\[
\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1}) \right|
\]
where \( (x_{n+1}, y_{n+1}) = (x_1, y_1) \).
So for our points: (0,0), (5,3), (7,0), (2,-3).
Plugging in:
\[
\text{Area} = \frac{1}{2} \left| 0*3 + 5*0 + 7*(-3) + 2*0 - (0*5 + 3*7 + 0*2 + (-3)*0)\right|
\]
\[
= \frac{1}{2} \left| 0 + 0 - 21 + 0 - (0 + 21 + 0 + 0) \right|
\]
\[
= \frac{1}{2} \left| -21 - 21 \right| = \frac{1}{2} \cdot 42 = 21
\]
### Conclusion
\[
\text{Area} = 21 \text{ square units}
\]
Therefore, the exact area of the rectangle based on the vertices is:
\[
\boxed{21.02 \text{ units squared}}
\]