Explanation
Part 1.
We graph the quadrilateral QUAD. For this, we plot and join the given ordered pairs.
As we can see, the quadrilateral QUAD is a kite because it has exactly 2 pairs of consecutive congruent sides.
Part 2.
Finding the perimeter
The perimeter is the sum of the length of all the sides of a polygon. Then, to calculate the perimeter of the quadrilateral QUAD, we have to find the measure of the segments UA and AD. For this, we can use the distance formula between two points.
[tex]\begin{gathered} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ \text{ Where }(x_1,y_1)\text{ and }(x_2,y_2)\text{ are the coordinates of the points.} \end{gathered}[/tex]
Then, we have:
• Measure of segment UA
[tex]\begin{gathered} (x_1,y_1)=U(0,6) \\ (x_2,y_2)=A(8,0) \\ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ d=\sqrt{(8-0)^2+(0-6)^2} \\ d=\sqrt{8^2+(-6)^2} \\ d=\sqrt{64+36} \\ d=\sqrt{100} \\ d=10 \end{gathered}[/tex]
• Measure of segment AD
[tex]\begin{gathered} (x_1,y_1)=A(8,0) \\ (x_2,y_2)=D(0,-15) \\ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ d=\sqrt{(0-8)^2+(-15-0)^2} \\ d=\sqrt{(-8)^2+(-15)^2} \\ d=\sqrt{64+225} \\ d=\sqrt{289} \\ d=17 \end{gathered}[/tex]
Now, we calculate the perimeter of the quadrilateral.
[tex]\begin{gathered} \text{ Perimeter }=\bar{UA}+\bar{AD}+\bar{DQ}+\bar{QU} \\ \text{ Perimeter }=10+17+17+10 \\ \text{ Perimeter }=54 \end{gathered}[/tex]
Therefore, the perimeter of QUAD is 54 units.
Finding the area
The area of a kite is half the product of the lengths of its diagonals.
[tex]A=\frac{1}{2}d_1*d_2[/tex]
Then, we have to calculate the measure of segments UD and QA to find the area of the kite. For this, we can use the distance formula between two points.
• Measure of segment UD
[tex]\begin{gathered} (x_1,y_1)=U(0,6) \\ (x_2,y_2)=D(0,-15) \\ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ d=\sqrt{(0-0)^2+(-15-6)^2} \\ d=\sqrt{0^2+(-21)^2} \\ d=\sqrt{0+21} \\ d=\sqrt{21} \\ d=21 \end{gathered}[/tex]
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