3. In the plane of a rectangular coordinate system XOY, construct a rectangle $ABCD$, given the coordinates of three of its vertices: $A(-3,-1), B(5,-1), C(5,3)$. Determine:

a) the coordinates of the fourth vertex,

b) the coordinates of the intersection of the lines $AC$ and $BD$.

c) the equations of the lines on which the sides and diagonals of this rectangle lie.
Solution. a) The coordinates of the fourth vertex $D$ are $(-3,4)$.

b) The coordinates of the intersection point $S$ of segments $A C$ and $B D$ are

$x_{S}=\frac{-3+5}{2}=1, y_{S}=\frac{-1+3}{2}=1$.

c) The equations of the lines on which the sides of the rectangle lie are

![](https://cdn.mathpix.com/cropped/2024_06_05_b58a8127523cd0cd5361g-3.jpg?height=592&width=718&top_left_y=1605&top_left_x=785)
$x=-3, x=5, y=-1, y=3$.

The equations of the lines on which the diagonals of the rectangle lie are:

$y+1=\frac{3+1}{5+3}(x+3), y+1=\frac{3+1}{-3-5}(x-5)$, i.e., $x-2 y+1=0, x+2 y-3=0$.