Question
Suppose a rectangle is drawn on a coordinate plane. What is the perimeter of the rectangle if its vertices are at coordinates (0,0), (4,4), (5,3), and (1,−1)? Round your answer to the nearest tenth.(1 point)
Responses
11.3 units
11.3 units
14.2 units
14.2 units
13.8 units
13.8 units
15.6 units
Responses
11.3 units
11.3 units
14.2 units
14.2 units
13.8 units
13.8 units
15.6 units
Answers
GPT-4o mini
To find the perimeter of the rectangle defined by the vertices \((0,0)\), \((4,4)\), \((5,3)\), and \((1,-1)\), we first need to verify if these points indeed form a rectangle.
A rectangle's diagonals must be equal in length and must bisect each other. We can calculate the lengths of the diagonals and the sides to determine if these points define a rectangle.
Let's calculate the lengths of the sides between consecutive points. We will use the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Calculating the distances between pairs of the given vertices:
1. Between \((0,0)\) and \((1,-1)\):
\[
d_1 = \sqrt{(1 - 0)^2 + (-1 - 0)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
\]
2. Between \((1,-1)\) and \((4,4)\):
\[
d_2 = \sqrt{(4 - 1)^2 + (4 - (-1))^2} = \sqrt{(3)^2 + (5)^2} = \sqrt{9 + 25} = \sqrt{34}
\]
3. Between \((4,4)\) and \((5,3)\):
\[
d_3 = \sqrt{(5 - 4)^2 + (3 - 4)^2} = \sqrt{(1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
\]
4. Between \((5,3)\) and \((0,0)\):
\[
d_4 = \sqrt{(0 - 5)^2 + (0 - 3)^2} = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}
\]
Now, let's identify the lengths that represent the sides of the rectangle. We note that the lengths \(\sqrt{2}\) and \(\sqrt{34}\) appear as opposites:
- Sides of length \(\sqrt{2}\) are the pairs \((0,0) \leftrightarrow (1,-1)\) and \((4,4) \leftrightarrow (5,3)\).
- Sides of length \(\sqrt{34}\) are the pairs \((1,-1) \leftrightarrow (4,4)\) and \((5,3) \leftrightarrow (0,0)\).
The rectangle’s perimeter \(P\) can be computed as follows:
\[
P = 2 \times (\text{length} + \text{width}) = 2 \times (\sqrt{34} + \sqrt{2})
\]
Calculating the lengths:
- \(\sqrt{34} \approx 5.831\)
- \(\sqrt{2} \approx 1.414\)
Then,
\[
P \approx 2 \times (5.831 + 1.414) = 2 \times 7.245 \approx 14.490
\]
Finally, rounding this to the nearest tenth:
\[
P \approx 14.5
\]
However, since the closest response is not available. We go by our closest quantitative calculation after reviewing the approximations.
Thus, the answer for the perimeter of the rectangle is:
**14.2 units**
A rectangle's diagonals must be equal in length and must bisect each other. We can calculate the lengths of the diagonals and the sides to determine if these points define a rectangle.
Let's calculate the lengths of the sides between consecutive points. We will use the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Calculating the distances between pairs of the given vertices:
1. Between \((0,0)\) and \((1,-1)\):
\[
d_1 = \sqrt{(1 - 0)^2 + (-1 - 0)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
\]
2. Between \((1,-1)\) and \((4,4)\):
\[
d_2 = \sqrt{(4 - 1)^2 + (4 - (-1))^2} = \sqrt{(3)^2 + (5)^2} = \sqrt{9 + 25} = \sqrt{34}
\]
3. Between \((4,4)\) and \((5,3)\):
\[
d_3 = \sqrt{(5 - 4)^2 + (3 - 4)^2} = \sqrt{(1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
\]
4. Between \((5,3)\) and \((0,0)\):
\[
d_4 = \sqrt{(0 - 5)^2 + (0 - 3)^2} = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}
\]
Now, let's identify the lengths that represent the sides of the rectangle. We note that the lengths \(\sqrt{2}\) and \(\sqrt{34}\) appear as opposites:
- Sides of length \(\sqrt{2}\) are the pairs \((0,0) \leftrightarrow (1,-1)\) and \((4,4) \leftrightarrow (5,3)\).
- Sides of length \(\sqrt{34}\) are the pairs \((1,-1) \leftrightarrow (4,4)\) and \((5,3) \leftrightarrow (0,0)\).
The rectangle’s perimeter \(P\) can be computed as follows:
\[
P = 2 \times (\text{length} + \text{width}) = 2 \times (\sqrt{34} + \sqrt{2})
\]
Calculating the lengths:
- \(\sqrt{34} \approx 5.831\)
- \(\sqrt{2} \approx 1.414\)
Then,
\[
P \approx 2 \times (5.831 + 1.414) = 2 \times 7.245 \approx 14.490
\]
Finally, rounding this to the nearest tenth:
\[
P \approx 14.5
\]
However, since the closest response is not available. We go by our closest quantitative calculation after reviewing the approximations.
Thus, the answer for the perimeter of the rectangle is:
**14.2 units**