Question

what is the best name for quadrilateral abcd given the points a(1,3), b(5,1), c(1, -1), and d(-3,1)? parallelogram rectangle rhombus square

Explanation:

Step1: Calculate side lengths

Use distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$:
$AB=\sqrt{(5-1)^2+(1-3)^2}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}$
$BC=\sqrt{(1-5)^2+(-1-1)^2}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}$
$CD=\sqrt{(-3-1)^2+(1-(-1))^2}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}$
$DA=\sqrt{(1-(-3))^2+(3-1)^2}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}$
All sides equal.

Step2: Check diagonals

$AC=\sqrt{(1-1)^2+(-1-3)^2}=\sqrt{0+16}=4$
$BD=\sqrt{(-3-5)^2+(1-1)^2}=\sqrt{64+0}=8$
Diagonals not equal.

Step3: Identify quadrilateral

Equal sides + unequal diagonals = rhombus.

Answer:

rhombus