Question

prove the diagonals of the square with vertices p(0, 4), q(4, 4), r(0, 0), and s(4, 0) are perpendicular bisectors of each other. step 1: calculate the slope of the diagonals. the slope of diagonal ps is dropdown. the slope of diagonal qr is dropdown. step 2: calculate the midpoint of the diagonals. the midpoint of ps is dropdown. the midpoint of qr is dropdown. the diagonals of the square are perpendicular bisectors because the diagonals are dropdown parallel and share the same midpoint perpendicular and share the same midpoint the same length

Explanation:

Step1: Calculate slope of PS

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $P(0,4)$ and $S(4,0)$, $m_{PS}=\frac{0 - 4}{4 - 0}=\frac{-4}{4}=- 1$.

Step2: Calculate slope of QR

For points $Q(4,4)$ and $R(0,0)$, $m_{QR}=\frac{0 - 4}{0 - 4}=\frac{-4}{-4}=1$.

Step3: Calculate mid - point of PS

The mid - point formula is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. For points $P(0,4)$ and $S(4,0)$, mid - point of $PS=(\frac{0 + 4}{2},\frac{4+0}{2})=(2,2)$.

Step4: Calculate mid - point of QR

For points $Q(4,4)$ and $R(0,0)$, mid - point of $QR=(\frac{4 + 0}{2},\frac{4+0}{2})=(2,2)$.

Answer:

The slope of diagonal PS is - 1.
The slope of diagonal QR is 1.
The midpoint of PS is (2,2).
The midpoint of QR is (2,2).
The diagonals of the square are perpendicular bisectors because the diagonals are perpendicular and share the same midpoint.