Question

rectangle abcd has the following vertices: a(-1,9) b(9,4) c(4,-6) d(-6,-1) is rectangle abcd a square, and why? choose 1 answer: a yes, because ab = bc = cd = ad, and abcd is a rectangle. b yes, because (overline{ab}paralleloverline{cd}) and (overline{bc}paralleloverline{ad}), and abcd is a rectangle. c no, because (overline{ab}) is longer than (overline{bc}).

Explanation:

Step1: Calculate the length of side AB using distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $A(-1,9)$ and $B(9,4)$, we have $AB=\sqrt{(9 + 1)^2+(4 - 9)^2}=\sqrt{100 + 25}=\sqrt{125}$.

Step2: Calculate the length of side BC using distance formula

For points $B(9,4)$ and $C(4,-6)$, we have $BC=\sqrt{(4 - 9)^2+(-6 - 4)^2}=\sqrt{25+100}=\sqrt{125}$.

Step3: Calculate the length of side CD using distance formula

For points $C(4,-6)$ and $D(-6,-1)$, we have $CD=\sqrt{(-6 - 4)^2+(-1+6)^2}=\sqrt{100 + 25}=\sqrt{125}$.

Step4: Calculate the length of side AD using distance formula

For points $A(-1,9)$ and $D(-6,-1)$, we have $AD=\sqrt{(-6 + 1)^2+(-1 - 9)^2}=\sqrt{25 + 100}=\sqrt{125}$.
Since $AB = BC = CD = AD$ and $ABCD$ is a rectangle.

Answer:

A. Yes, because $AB = BC = CD = AD$, and $ABCD$ is a rectangle.