Question
what is the area of a rectangle if the coordinates of it's points are; (-3,-1), (3,1), (4,-2), and (-2,-4)
Answers
GPT-4o mini
To find the area of a rectangle given its vertices, we need to first identify which pairs of points form the opposite corners of the rectangle.
The coordinates of the four points given are:
1. \( A(-3, -1) \)
2. \( B(3, 1) \)
3. \( C(4, -2) \)
4. \( D(-2, -4) \)
We can check which points are diagonal corners of the rectangle by calculating the distances between all pairs of points and identifying the length and width of the rectangle.
Calculating the distances, we find:
1. \( AB = \sqrt{(3 - (-3))^2 + (1 - (-1))^2} = \sqrt{(6)^2 + (2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \)
2. \( AC = \sqrt{(4 - (-3))^2 + (-2 - (-1))^2} = \sqrt{(7)^2 + (-1)^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2} \)
3. \( AD = \sqrt{(-2 - (-3))^2 + (-4 - (-1))^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \)
4. \( BC = \sqrt{(4 - 3)^2 + (-2 - 1)^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \)
5. \( BD = \sqrt{(-2 - 3)^2 + (-4 - 1)^2} = \sqrt{(-5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \)
6. \( CD = \sqrt{(-2 - 4)^2 + (-4 - (-2))^2} = \sqrt{(-6)^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \)
Based on the distances calculated, the opposite corners seem to be the pairs:
- \( A(-3, -1) \) and \( C(4, -2) \) as one diaogonal pair.
- \( B(3, 1) \) and \( D(-2, -4) \) as the other diagonal pair.
To find the area of the rectangle, we can use the distance formula to find the length of the sides formed by these points:
- Length (from \( A \) to \( B \) or \( C \) to \( D \)):
\[
\text{Length} = |x_C - x_A| = |4 - (-3)| = |4 + 3| = 7
\]
- Width (from \( A \) to \( D \) or \( B \) to \( C \)):
\[
\text{Width} = |y_B - y_A| = |1 - (-1)| = |1 + 1| = 2
\]
Now, we can calculate the area of the rectangle:
\[
\text{Area} = \text{Length} \times \text{Width} = 7 \times 2 = 14
\]
Thus, the area of the rectangle is \( \boxed{14} \).
The coordinates of the four points given are:
1. \( A(-3, -1) \)
2. \( B(3, 1) \)
3. \( C(4, -2) \)
4. \( D(-2, -4) \)
We can check which points are diagonal corners of the rectangle by calculating the distances between all pairs of points and identifying the length and width of the rectangle.
Calculating the distances, we find:
1. \( AB = \sqrt{(3 - (-3))^2 + (1 - (-1))^2} = \sqrt{(6)^2 + (2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \)
2. \( AC = \sqrt{(4 - (-3))^2 + (-2 - (-1))^2} = \sqrt{(7)^2 + (-1)^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2} \)
3. \( AD = \sqrt{(-2 - (-3))^2 + (-4 - (-1))^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \)
4. \( BC = \sqrt{(4 - 3)^2 + (-2 - 1)^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \)
5. \( BD = \sqrt{(-2 - 3)^2 + (-4 - 1)^2} = \sqrt{(-5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \)
6. \( CD = \sqrt{(-2 - 4)^2 + (-4 - (-2))^2} = \sqrt{(-6)^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \)
Based on the distances calculated, the opposite corners seem to be the pairs:
- \( A(-3, -1) \) and \( C(4, -2) \) as one diaogonal pair.
- \( B(3, 1) \) and \( D(-2, -4) \) as the other diagonal pair.
To find the area of the rectangle, we can use the distance formula to find the length of the sides formed by these points:
- Length (from \( A \) to \( B \) or \( C \) to \( D \)):
\[
\text{Length} = |x_C - x_A| = |4 - (-3)| = |4 + 3| = 7
\]
- Width (from \( A \) to \( D \) or \( B \) to \( C \)):
\[
\text{Width} = |y_B - y_A| = |1 - (-1)| = |1 + 1| = 2
\]
Now, we can calculate the area of the rectangle:
\[
\text{Area} = \text{Length} \times \text{Width} = 7 \times 2 = 14
\]
Thus, the area of the rectangle is \( \boxed{14} \).