Given:
A(1,2)
B(3,4)
C(3,-1)
D(1,-3)
Solution:
We are asked to justify the argument using the slope formula and the distance formula.
Slope Formula:
[tex]m=\frac{y_2-y_1}{x_2-x_{`1}}[/tex]
Distance Formula:
[tex]d=\sqrt[]{(x_2_{}-x_1)^2^{}_{}+(y_2-y_1_{})^2}[/tex]
Now, we will use the given to find the slope and the distance inorder to answer the questions.
For the slopes:
[tex]\begin{gathered} m_{AB}=\frac{4-2}{3-1} \\ m_{AB}=\frac{2}{2} \\ m_{AB}=1 \end{gathered}[/tex]
[tex]\begin{gathered} m_{CD}=\frac{-3+1}{1-3} \\ m_{CD}=\frac{-2}{-2} \\ m_{CD}=1 \end{gathered}[/tex]
The slopes of AB and CD are equal.
For the distances:
[tex]\begin{gathered} d_{AB}=\sqrt[]{(3_{}-1_{})^2_{}+(4_{}-2_{})^2} \\ d_{AB}=\sqrt[]{2^2+2^2} \\ d_{AB}=\sqrt[]{8} \\ d_{AB}=2\sqrt[]{2} \end{gathered}[/tex]
[tex]\begin{gathered} d_{CD}=\sqrt[]{(1_{}-3_{})^2_{}+(-3_{}+1_{})^2} \\ d_{CD}=\sqrt[]{(-2)^2+(-2)^2} \\ d_{CD}=\sqrt[]{8} \\ d_{CD}=2\sqrt[]{2} \end{gathered}[/tex]
ANSWER:
Yes, ABCD is a paralellogram.
AB is equal to CD and the slopes of AB and CD are equal, so one pair of opposite sides is both parallel and congruent.
