Question

find the slopes of each side of the quadrilateral to prove that dkmp is a parallelogram using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.
slope of pd =
slope of mk =
slope of pm =
slope of dk =

Explanation:

Step1: Find slope of PD

Points \( P(-6, 2) \) and \( D(0, 6) \). Using slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \), substitute \( y_2 = 6, y_1 = 2, x_2 = 0, x_1 = -6 \).
\( m_{PD}=\frac{6 - 2}{0 - (-6)}=\frac{4}{6}=\frac{2}{3} \)

Step2: Find slope of MK

Points \( M(-5, -3) \) and \( K(1, 1) \). Substitute \( y_2 = 1, y_1 = -3, x_2 = 1, x_1 = -5 \) into slope formula.
\( m_{MK}=\frac{1 - (-3)}{1 - (-5)}=\frac{4}{6}=\frac{2}{3} \)

Step3: Find slope of PM

Points \( P(-6, 2) \) and \( M(-5, -3) \). Substitute \( y_2 = -3, y_1 = 2, x_2 = -5, x_1 = -6 \) into slope formula.
\( m_{PM}=\frac{-3 - 2}{-5 - (-6)}=\frac{-5}{1}=-5 \)

Step4: Find slope of DK

Points \( D(0, 6) \) and \( K(1, 1) \). Substitute \( y_2 = 1, y_1 = 6, x_2 = 1, x_1 = 0 \) into slope formula.
\( m_{DK}=\frac{1 - 6}{1 - 0}=\frac{-5}{1}=-5 \)

Answer:

Slope of \( PD = \boldsymbol{\frac{2}{3}} \)
Slope of \( MK = \boldsymbol{\frac{2}{3}} \)
Slope of \( PM = \boldsymbol{-5} \)
Slope of \( DK = \boldsymbol{-5} \)

(Note: Since \( m_{PD}=m_{MK} \) and \( m_{PM}=m_{DK} \), opposite sides are parallel, so \( DKMP \) is a parallelogram.)