The coordinates of three vertices of a rectangle are shown on the grid.
A coordinate graph with x and y axes labeled from negative 5 to 5 is shown with a black dots at the points negative 1 comma negative 2, negative 1 comma 4, and 3 comma 4.
Which could be the coordinates of the fourth vertex?
A. (3, –2)
B. (3, –1)
C. (–2, 3)
D. (4, –2)
Since the rectangle has perpendicular sides, we can use the midpoint formula. The midpoint of $(-1,-2)$ and $(-1,4)$ is $(-1,1)$. The midpoint of $(-1,-2)$ and $(3,4)$ is $(1,1)$.
[asy]
size(100);
draw((-5,0)--(5,0),Arrows);
draw((0,-5)--(0,5),Arrows);
dot((-1,-2));
dot((-1,4));
dot((3,4));
label("(-1,-2)",(-1,-2),SW);
label("(-1,4)",(-1,4),W);
label("(3,4)",(3,4),NE);
draw((-1,-2)--(-1,4)--(3,4));
label("(-1,1)",(-1,1),W);
label("(1,1)",(1,1),NE);
[/asy]
The midpoint of $(-1,1)$ and $(3,4)$ is $(1.0,2.5)$. Since the $y$-coordinate is increasing, we can eliminate choices $\textbf{(A)}$ and $\textbf{(B)}$. The point $(1,1)$ is the midpoint of horizontal side of length 4, so the fourth vertex must be the midpoint of the vertical side of length 4.
Therefore, the fourth vertex is $(1,-2)$, which is $\boxed{\textbf{(D)}\ (4,-2)}$ mirrored across the line $y=-1$.
can u put this in baby reading-
We want to find the missing point in a rectangle given three other points. We use a trick called the midpoint formula to figure out that the missing point is at (1,-2). The only answer choice that contains (1,-2) is $\boxed{\textbf{(D)}\ (4,-2)}$.
To find the coordinates of the fourth vertex of a rectangle, we can use the properties of rectangles.
In a rectangle, opposite sides are parallel and equal in length. Additionally, the diagonals of a rectangle bisect each other.
In this case, we have three given vertices: (-1, -2), (-1, 4), and (3, 4).
To find the possible coordinates of the fourth vertex, we need to consider the properties of a rectangle.
1. Opposite sides of a rectangle are parallel and equal in length.
The length between (-1, -2) and (-1, 4) is 6 units (4 - (-2) = 6). Therefore, the length between (-1, -2) and (3, 4) should also be 6 units.
The length between (-1, -2) and (3, 4) can be calculated as follows:
Length = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((3 - (-1))^2 + (4 - (-2))^2)
= sqrt((4)^2 + (6)^2)
= sqrt(16 + 36)
= sqrt(52)
The distance between (-1, -2) and (3, 4) is sqrt(52), which is not equal to 6.
So, option A (3, -2) cannot be the coordinates of the fourth vertex.
2. The diagonals of a rectangle bisect each other.
The midpoint between (-1, -2) and (3, 4) can be calculated as:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
= ((-1 + 3) / 2, (-2 + 4) / 2)
= (1 / 2, 2 / 2)
= (1 / 2, 1)
The midpoint between (-1, -2) and (3, 4) is (1/2, 1).
Now, let's consider the given options:
A. (3, –2):
The y-coordinate does not match the midpoint (1/2, 1).
B. (3, –1):
The y-coordinate does not match the midpoint (1/2, 1).
C. (–2, 3):
The midpoint (1/2, 1) does not lie on the line segment formed by (-1, -2) and (3, 4).
D. (4, –2):
The y-coordinate does not match the midpoint (1/2, 1).
From the options given, none of them has the correct coordinates for the fourth vertex.