Properties of Quadrilaterals

In order to determine the midlength, we must first calculate the side lengths of \(\text{WX}\) and \(\text{ZY}\):

\(d = \sqrt{(x^2 - x^1)^2 + (y^2 - y^1 )^2}\)

First, we can calculate the side length of \(\text{WX}\):

\(d_{{\text{WX}}} = \sqrt{(3 - 1)^2 + (1 - 2)^2}\)

\(d_{{\text{WX}}} = \sqrt{(2)^2 + (-1)^2}\)

\(d_{{\text{WX}}} = \sqrt{4 + 1} = \sqrt{5}\)

\(d_{{\text{WX}}} = \sqrt{5}\)

Next, we can calculate the side length of \(\text{ZY}\):

\(d_{{\text{ZY}}} = \sqrt{(2 - (-4))^2 + (-4 - (-1))^2}\)

\(d_{{\text{ZY}}} = \sqrt{(6)^2 + (-3)^2}\)

\(d_{{\text{ZY}}} = \sqrt{36 + 9}\)

\(d_{{\text{ZY}}} = \sqrt{45}\)

In order to determine the length of the midsegment, all we need to do is take the average of the \(2\) parallel sides:

\(d_{\text{midsegment}} = \cfrac{d_{{\text{ZY}}} + d_{{\text{WX}}}}{2}\)

\(d_{\text{midsegment}} = \cfrac{\sqrt{45} + \sqrt{5}}{2}\)

\(d_{\text{midsegment}} = \cfrac{6.71 + 2.23}{2}\)

\(d_{\text{midsegment}} = \cfrac{8.94}{2}\)

\(d_{\text{midsegment}} \approx 4.47\)

Therefore, we can determine that the length of the midsegment is approximately \(\boldsymbol{4.47}\).

i. We can use the midpoint formula to find the midpoints of each of the respective side lengths:

\(\text{midpoint} = \left(\cfrac{x_1 + x_2}{2}, \cfrac{y_1 + y_2}{2}\right)\)

First, we can determine the midpoint for Point \(\text{D}\):

\(\text{midpoint}_{\text{D}} = \left(\cfrac{-2 - 4}{2}, \cfrac{4 - 2}{2}\right) = \left(\cfrac{-6}{2}, \cfrac{2}{2}\right) = (-3, 1)\)

Next, we can determine the midpoint for Point \(\text{E}\):

\(\text{midpoint}_{\text{E}} = \left(\cfrac{-4 + 2}{2}, \cfrac{-2 - 4}{2}\right) = \left(\cfrac{-2}{2}, \cfrac{-6}{2}\right) = (-1, -3)\)

Then, we can determine the midpoint for Point \(\text{F}\):

\(\text{midpoint}_{\text{F}} = \left(\cfrac{2 + 4}{2}, \cfrac{-4 + 0}{2}\right) = \left(\cfrac{6}{2}, \cfrac{-4}{2}\right) = (3, -2)\)

Finally, we can determine the midpoint for Point \(\text{G}\):

\(\text{midpoint}_{\text{G}} = \left(\cfrac{-2 + 4}{2}, \cfrac{4 + 0}{2}\right) = \left(\cfrac{2}{2}, \cfrac{4}{2}\right) = (1, 2)\)

Therefore, we can determine that the respecive midpoints are \(\textbf{D(-3,1)}\), \(\textbf{E(-1,-3)}\), \(\textbf{F(3,-2)}\) and \(\textbf{G(1,2)}\).

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ii. To find that the opposite sides are parallel, we can find the respective slopes for each side length. In this instance, \(\text{DE}\) is opposite to \(\text{FG}\) and \(\text{EF}\) is opposite to \(\text{DG}\):

\(m = \cfrac{y_2-y_1}{x_2-x_1}\)

First, we can find the slope of line \(\text{DE}\):

\(m_{(\text{DE})} = \cfrac{-3 - 1}{-1 - (-3)} = \cfrac{-4}{2} = -2\)

Next, we can find the slope of line \(\text{EF}\):

\(m_{\text{EF}} = \cfrac{-2 - (-3)}{3 - (-1)} = \cfrac{1}{4}\)

Then, we can find the slope of line \(\text{FG}\):

\(m_{\text{FG}} = \cfrac{2 - (-2)}{1 - 3} = \cfrac{4}{-2} = -2\)

Finally, we can find the slope of line \(\text{DG}\):

\(m_{\text{DG}} = \cfrac{1 - 2}{-3 - 1} = \cfrac{-1}{-4} = \cfrac{1}{4}\)

As the slopes of \(\text{DE} = \text{FG}\) and \(\text{EF} = \text{DG}\), we can determine that the opposite sides are parallel to each other.

We can verify that the opposite sides are equal in length by using the length of a line formula:

\(d = \sqrt{(x^2 - x^1)^2 + (y^2 - y^1 )^2}\)

First, we can determine the side length of line \(\text{DE}\):

\(d_{\text{DE}} = \sqrt{(-1 - (-3))^2 + (-3 - 1)^2} = \sqrt{(2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20}\)

Next, we can determine the side length of line \(\text{EF}\):

\(d_{\text{EF}} = \sqrt{(3 - (-1))^2 + (-2 - (-3))^2} = \sqrt{(4)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{17}\)

Then, we can determine the side length of line \(\text{FG}\):

\(d_{\text{FG}} = \sqrt{(1 - 3)^2 + (2 - (-2))^2} = \sqrt{(-2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20}\)

Finally, we can determine the side length of line \(\text{DG}\):

\(d_{\text{DG}} = \sqrt{(1 - (-3))^2 + (2 - 1)^2} = \sqrt{(4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17}\)

As the side lengths of \(\text{DE} = \text{FG}\) and \(\text{EF} = \text{DG}\), we can determine that the opposite sides are equal in length.

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iii. We can determine that \(\text{DEFG}\) is a parallelogram since none of the slopes are negative reciprocols.

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iv. Based on the coordinates we have determined from the previous questions, we can graph both \(\text{STUV}\) and \(\text{DEFG}\):

i. In order to verify that quadrilateral \(\text{ABCD}\) is a rhombus, we can calculate the side lengths of each side:

\(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)

First, we can find the side length of \(\text{AB}\):

\(d_{\text{AB}} = \sqrt{(5-2)^2 + (6-2)^2} = \sqrt{(3)^2 + (4)^2} = \sqrt{9+16} = \sqrt{25} = 5\)

Next, we can find the side length of \(\text{BC}\):

\(d_{\text{BC}} = \sqrt{(10-5)^2 + (6-6)^2} = \sqrt{(5)^2 + (0)^2} = \sqrt{25} = 5\)

Then, we can find the side length of \(\text{CD}\):

\(d_{\text{CD}} = \sqrt{(7-10)^2 + (2-6)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)

Finally, we can find the side length of \(\text{AD}\):

\(d_{\text{AD}} = \sqrt{(7-2)^2 + (2-2)^2} = \sqrt{(5)^2 + (0)^2} = \sqrt{25} = 5\)

As the side lengths of \(\text{AB} = \text{BC} = \text{CD} = \text{AD}\), we can verify that quadrilateral \(\text{ABCD}\) represents a rhombus.

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ii. If the diagonals have the same midpoint, they bisect each other.

We can find the coordinates of the midpoint for each diagonal:

\(\text{midpoint} = \left(\cfrac{x_1 + x_2}{2}, \cfrac{y_1 + y_2}{2}\right)\)

First, we can determine the midpoint of the diagonal \(\text{AC}\):

\(\text{midpoint}_{\text{AC}} = \left(\cfrac{2+10}{2}, \cfrac{2+6}{2}\right) = \left(\cfrac{12}{2}, \cfrac{8}{2}\right) = (6, 4)\)

Next, we can determine the midpoint of the diagonal \(\text{BD}\):

\(\text{midpoint}_{\text{BD}} = \left(\cfrac{5+7}{2}, \cfrac{6+2}{2}\right) = \left(\cfrac{12}{2}, \cfrac{8}{2}\right) = (6, 4)\)

Since the midpoints of the diagonals have the same coordinates, we can verify the diagonals bisect each other.