The coordinates of a quadrilateral are (1,3), (7,-3), (1,-9), and (-5,-3). which shape is formed select all that apply -kite -rectangle -rhombus -square -trapezoid

rectangle , rhombus , and square

Step-by-step explanation:

First we find the slope of the line between each pair of points.  We will first name each point:

A(1, 3); B(7, -3); C(1, -9); and D(-5, -3).

The formula for slope is

This means the slope for AB is

m = (3--3)/(1-7) = (3+3)/(-6) = 6/-6 = -1

The slope for BC is

m = (-3--9)/(7-1) = (-3+9)/6 = 6/6 = 1

The slope for CD is

m = (-9--3)/(1--5) = (-9+3)/(1+5) = (-6)/6 = -1

The slope for DA is

m = (-3-3)/(-5-1) = (-6)/(-6) = 1

If two sides are parallel, then the slopes of their lines are the same.  The slopes of AB and CD are the same; this means they are parallel.  The slopes of BC and DA are the same; this means they are parallel.  This makes this figure a parallelogram.

If two sides form a right angle, then their slopes are negative reciprocals.  The slopes of AB and BC are negative reciprocals, so they form a right angle.  The slopes of BC and CD are negative reciprocals, so they form a right angle.  The slopes of CD and AB are negative reciprocals, so they form a right angle.  This means the fourth angle must be a right angle as well.  This makes the figure a rectangle.

We next use the distance formula to find the length of each side:

The length of AB is

The length of BC is

The length of CD is

The length of DA is

Since all four sides have the same length, the figure is a square.