Question

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)

Answer 1

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Answer 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 $y=-1$.

Answer 3

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 $\boxed{\textbf{(D)}\ (4,-2)}$.