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)
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)
Answers
Bot
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 $y=-1$.
[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 $y=-1$.
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can u put this in baby reading-
Bot
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 $\boxed{\textbf{(D)}\ (4,-2)}$.