Question

select the correct location on the table. diana and sean are considering the quadrilateral with coordinates q(-2,4), r(1,5), s(2,2), and t(-1,1). they already determined the sides of the quadrilateral are perpendicular and agree that qrst is a rectangle. however, sean believes qrst is also a square. which set of work correctly determines whether sean is correct?

Explanation:

Step1: Calculate the length of side QR

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $Q(-2,4)$ and $R(1,5)$, we have $x_1=-2,y_1 = 4,x_2=1,y_2 = 5$. Then $QR=\sqrt{(1-(-2))^2+(5 - 4)^2}=\sqrt{(3)^2+(1)^2}=\sqrt{9 + 1}=\sqrt{10}$.

Step2: Calculate the length of side RS

For points $R(1,5)$ and $S(2,2)$, $x_1=1,y_1 = 5,x_2=2,y_2 = 2$. Then $RS=\sqrt{(2 - 1)^2+(2 - 5)^2}=\sqrt{(1)^2+(-3)^2}=\sqrt{1+9}=\sqrt{10}$.

Step3: Calculate the length of side ST

For points $S(2,2)$ and $T(-1,1)$, $x_1=2,y_1 = 2,x_2=-1,y_2 = 1$. Then $ST=\sqrt{(-1 - 2)^2+(1 - 2)^2}=\sqrt{(-3)^2+(-1)^2}=\sqrt{9 + 1}=\sqrt{10}$.

Step4: Calculate the length of side TQ

For points $T(-1,1)$ and $Q(-2,4)$, $x_1=-1,y_1 = 1,x_2=-2,y_2 = 4$. Then $TQ=\sqrt{(-2-(-1))^2+(4 - 1)^2}=\sqrt{(-1)^2+(3)^2}=\sqrt{1 + 9}=\sqrt{10}$.
Since all four - side lengths are equal ($QR = RS=ST=TQ=\sqrt{10}$) and the sides are perpendicular (already known), the rectangle QRST is a square.

Answer:

The set of work that calculates the lengths of all four sides using the distance formula and shows that all four side - lengths are equal correctly determines that Sean is correct.