Question

Use the figure above of Quadrilateral ABCD to answer the following: Use the check list to determine which properties fit. You will need to find the distance and slope of AB, BC, CD, and AD. option 1 : Opposite sides (AB, CD and BC, AD) have equal slopes making the parallel to each other, making ABCD a parallelogram.option 2: There is only two sides that have equal slopes, making them parallel to each otheroption 3: The slope of adjacent sides (like AB and BC) have negative reciprocals for slopes, making them perpendicular, which makes four 90 degree anglesoption 4: The distance of the sides AB, BC, CD, and AD are all congruent option 5 : The distance of the sides AB and CD are congruent and BC and AD are congruent, making opposite sides congruent, but not all four sides

Answer 1

Step-by-step explanation:

First, let's draw the quadrilateral. So:

Then, the distance d and slope m between two points with coordinates (x1, y1) and (x2, y2) can be calculated as:

`\begin{gathered} d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ m=(y_2-y_1)/(x_2-x_1) \end{gathered}`

So, the distance and slope of AB where A is (-5,3) and B is (0, 6) are:

`\begin{gathered} d=\sqrt[]{(0-(-5))^2+(6-3)^2} \\ d=\sqrt[]{(0+5)^2+3^2} \\ d=\sqrt[]{34} \\ m=(6-3)/(0-(-5))=\frac{3}{0+5_{}}=(3)/(5) \end{gathered}`

The distance and slope of BC where B is (0,6) and C is (5, 3) are:

`\begin{gathered} d=\sqrt[]{(5-0)^2+(3-6)^2}=\sqrt[]{34} \\ m=(3-6)/(5-0)=-(3)/(5) \end{gathered}`

The distance and slope of CD where C is (5,3) and D is (0, 0) is:

`\begin{gathered} d=\sqrt[]{(0-5)^2+(0-3)^2}=\sqrt[]{34} \\ m=(0-3)/(0-5)=(-3)/(-5)=(3)/(5) \end{gathered}`

The distance and slope of AD where A is (-5,3) and D is (0, 0) are:

`\begin{gathered} d=\sqrt[]{(0-(-5))^2+(0-3)^2} \\ d=\sqrt[]{(0+5)^2+(3)^2}=\sqrt[]{34} \\ m=(0-3)/(0-(-5))=-(3)/(5) \end{gathered}`

Therefore, the correct answers are:

Option 1 : Opposite sides (AB, CD, and BC, AD) have equal slopes making them parallel to each other, making ABCD a parallelogram.

Option 4: The distance of the sides AB, BC, CD, and AD are all congruent