Answer:
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
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
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:
[tex]d = \sqrt{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]
The length of AB is
[tex]d=\sqrt{(3--3)^2+(1-7)^2}=\sqrt{6^2+(-6)^2}=\sqrt{36+36}=\sqrt{72}[/tex]
The length of BC is
[tex]d=\sqrt{(-3--9)^2+(7-1)^2}=\sqrt{6^2+6^2}=\sqrt{36+36}=\sqrt{72}[/tex]
The length of CD is
[tex]d=\sqrt{(-9--3)^2+(1--5)^2}=\sqrt{(-6)^2+6^2}=\sqrt{36+36}=\sqrt{72}[/tex]
The length of DA is
[tex]d=\sqrt{(-3-3)^2+(-5-1)^2}=\sqrt{(-6)^2+(-6)^2}=\sqrt{36+36}=\sqrt{72}[/tex]
Since all four sides have the same length, the figure is a square.
Answer: rectangle , rhombus , and square
Step-by-step explanation:
i got it right :)
