ABCD is a rectangle. The coordinates of point A are $(3, 5)$. The midpoint of AB, point M, has coordinates $(1, 5)$. The midpoint of BC, point N, has coordinates $(-1, -1)$. Find the coordinates of point D.

1. Problem Description

ABCD is a rectangle. The coordinates of point A are $(3, 5)$. The midpoint of AB, point M, has coordinates $(1, 5)$. The midpoint of BC, point N, has coordinates $(-1, -1)$. Find the coordinates of point D.

2. Solution Steps

First, let's find the coordinates of point B using the midpoint formula. The midpoint M of a line segment with endpoints A $(x_1, y_1)$ and B $(x_2, y_2)$ is given by:

$M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$

We know that M is $(1, 5)$ and A is $(3, 5)$. Let B be $(x_B, y_B)$. Then:

$(1, 5) = (\frac{3 + x_B}{2}, \frac{5 + y_B}{2})$

Equating the x-coordinates:

$1 = \frac{3 + x_B}{2}$

$2 = 3 + x_B$

$x_B = -1$

Equating the y-coordinates:

$5 = \frac{5 + y_B}{2}$

$10 = 5 + y_B$

$y_B = 5$

So, the coordinates of point B are $(-1, 5)$.

Next, let's find the coordinates of point C using the midpoint formula. The midpoint N of a line segment with endpoints B $(x_1, y_1)$ and C $(x_2, y_2)$ is given by:

$N = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$

We know that N is $(-1, -1)$ and B is $(-1, 5)$. Let C be $(x_C, y_C)$. Then:

$(-1, -1) = (\frac{-1 + x_C}{2}, \frac{5 + y_C}{2})$

Equating the x-coordinates:

$-1 = \frac{-1 + x_C}{2}$

$-2 = -1 + x_C$

$x_C = -1$

Equating the y-coordinates:

$-1 = \frac{5 + y_C}{2}$

$-2 = 5 + y_C$

$y_C = -7$

So, the coordinates of point C are $(-1, -7)$.

Finally, since ABCD is a rectangle, we know that $\vec{AD} = \vec{BC}$.

Let D be $(x_D, y_D)$. Also, let A be $(x_A, y_A)$, B be $(x_B, y_B)$, and C be $(x_C, y_C)$. Then

$(x_D - x_A, y_D - y_A) = (x_C - x_B, y_C - y_B)$

$(x_D - 3, y_D - 5) = (-1 - (-1), -7 - 5)$

$(x_D - 3, y_D - 5) = (0, -12)$

Equating the x-coordinates:

$x_D - 3 = 0$

$x_D = 3$

Equating the y-coordinates:

$y_D - 5 = -12$

$y_D = -7$

Thus, the coordinates of point D are $(3, -7)$.

3. Final Answer

The coordinates of point D are $(3, -7)$.