The vertex of a rectangle whose side lengths are 6 and 8 lies at the point (-4,6) in the rectangular coordinate plane. Which of the following points could be a vertex of this rectangle?
(A) (2,-2)
(B) (2,8)
(C) (-4,-10)
(D) (-4,-4)
(E) (-10,2)
-4±6 = 2 or -10
6±8 = 14 or -2
-4±8 = 4 or -12
6±6 = 12 or 0
So, pick the point which satisfies those conditions.
Looks like (A) to me
To determine which of the given points could be a vertex of the rectangle, we can use the fact that opposite sides of a rectangle are equal in length.
Given that one side length is 6, the opposite side length must also be 6. Similarly, one side length is 8, so the opposite side length must also be 8.
Let's find the possible points:
(A) (2, -2): This point is not a possible vertex because the length of the vertical side is 8, not 6.
(B) (2, 8): This point is not a possible vertex because the length of the horizontal side is 6, not 8.
(C) (-4, -10): This point is not a possible vertex because the length of the vertical side is 8, not 6.
(D) (-4, -4): This point is a possible vertex since the length of the horizontal side is 6 and the length of the vertical side is 8.
(E) (-10, 2): This point is not a possible vertex because the length of the horizontal side is 6, not 8.
Therefore, the answer is (D) (-4, -4).
To determine which of the points could be a vertex of the rectangle, we need to understand the properties of rectangles and analyze the given information.
1. The vertex of a rectangle is the point where two adjacent sides meet. In this case, the given vertex is (-4,6).
2. The side lengths of the rectangle are given as 6 and 8. This means that the adjacent sides meeting at the vertex will have lengths of 6 and 8.
3. Since the rectangle is aligned with the rectangular coordinate plane, the sides will be parallel to the x and y axes.
Now, let's analyze each given point:
(A) (2,-2)
- The distance between this point and the given vertex is sqrt((2 - (-4))^2 + (-2 - 6)^2) = sqrt(36 + 64) = sqrt(100) = 10.
- The difference between the side lengths of the rectangle is 8 - 6 = 2.
(B) (2,8)
- The distance between this point and the given vertex is sqrt((2 - (-4))^2 + (8 - 6)^2) = sqrt(36 + 4) = sqrt(40) which is not an integer.
- The difference between the side lengths of the rectangle is 8 - 6 = 2.
(C) (-4,-10)
- The distance between this point and the given vertex is sqrt((-4 - (-4))^2 + (-10 - 6)^2) = sqrt(0 + 256) = sqrt(256) = 16.
- The difference between the side lengths of the rectangle is 8 - 6 = 2.
(D) (-4,-4)
- The distance between this point and the given vertex is sqrt((-4 - (-4))^2 + (-4 - 6)^2) = sqrt(0 + 100) = sqrt(100) = 10.
- The difference between the side lengths of the rectangle is 8 - 6 = 2.
(E) (-10,2)
- The distance between this point and the given vertex is sqrt((-10 - (-4))^2 + (2 - 6)^2) = sqrt(36 + 16) = sqrt(52) which is not an integer.
- The difference between the side lengths of the rectangle is 8 - 6 = 2.
Based on the calculations, the only point with an integral distance from the given vertex and a difference in the side lengths of 2 is (D) (-4,-4).
Therefore, the correct answer is (D) (-4,-4).