What is area of the rectangle
A= -1,4
B= 5,2
C= -4,-5
D= 2,-7
Possible answers are
20√13
60
10√6
120
Please help
To find the area of the rectangle formed by the points A, B, C, and D, we can use the distance formula to find the lengths of the sides.
First, find the lengths of the sides AB, BC, CD, and DA:
AB = √[(5 - (-1))^2 + (2 - 4)^2] = √[6^2 + (-2)^2] = √(36 + 4) = √40 = 2√10
BC = √[(5 - (-4))^2 + (2 - (-5))^2] = √[9^2 + 7^2] = √(81 + 49) = √130
CD = √[(-4 - 2)^2 + (-5 - 7)^2] = √[(-6)^2 + (-12)^2] = √(36 + 144) = √180 = 6√5
DA = √[(-4 - (-1))^2 + (-5 - 2)^2] = √[(-3)^2 + (-7)^2] = √(9 + 49) = √58
Now, we can see that AB and CD are the diagonals of the rectangle. The area of a rectangle can be found by multiplying the lengths of the diagonals and dividing by 2:
Area = (AB * CD) / 2
Area = (2√10 * 6√5) / 2
Area = 12√50 / 2
Area = 6√50
Area = 6 * √(25*2)
Area = 6 * 5√2
Area = 30√2
Therefore, the area of the rectangle formed by the points A, B, C, and D is 30√2. Unfortunately, none of the provided answers match this result.
I think I'm having trouble getting the right calculations because it's a diagonal rectangle I wish I could and you a picture
I understand that visualizing a diagonal rectangle can be a bit challenging without a picture. Let's try a different approach to calculate the area of the rectangle using the coordinates of the vertices A, B, C, and D.
To find the area of the rectangle, we can use the distance formula to find the lengths of two adjacent sides and then multiply them to get the area.
Let's calculate the lengths of the sides using the given coordinates:
1. Length of side AB:
$$
AB = √[(5 - (-1))^2 + (2 - 4)^2] \\
AB = √[6^2 + (-2)^2] \\
AB = √(36 + 4) \\
AB = √40 = 2√10
$$
2. Length of side BC:
$$
BC = √[(5 - (-4))^2 + (2 - (-5))^2] \\
BC = √[9^2 + 7^2] \\
BC = √(81 + 49) \\
BC = √130
$$
The area of a rectangle is given by the formula: Area = Length x Width
In a rectangle, opposite sides are equal. Therefore, the length and width of the rectangle are AB and BC.
Area = AB x BC = 2√10 * √130 = 2√(10 * 130) = 2√(1300) = 20√13
Therefore, the area of the rectangle formed by the points A, B, C, and D is 20√13, which matches one of the provided answers.