The given quadrilateral is a kite.
Given: Point A (2, 4), B (-2, -5), C (7, -1) and D (7, 4)
Firstly, we find the distance between AD and DC
AD = [tex]\sqrt{(7 - 2)^{2} + (4 - 4)^{2} }[/tex]
⇒ AD = [tex]\sqrt{5^{2} }[/tex]
⇒ AD = 5
DC = [tex]\sqrt{(7 - 7)^{2} + (4 - (-1))^{2} }[/tex]
⇒ DC = [tex]\sqrt{5^{2} }[/tex]
⇒ DC = 5
Hence, AD = DC = 5
Now, find the distance between AB and BC
AB = [tex]\sqrt{(-2 - 2)^{2} + (-5 - 4)^{2} }[/tex]
⇒ AB = [tex]\sqrt{(-4)^{2} + (-9)^{2} }[/tex]
⇒ AB = [tex]\sqrt{16 + 81}[/tex]
⇒ AB = [tex]\sqrt{97}[/tex]
BC = [tex]\sqrt{(7 - (-2))^{2} + (-1 - (-5))^{2} }[/tex]
⇒ BC = [tex]\sqrt{9^{2} + 4^{2} }[/tex]
⇒ BC = [tex]\sqrt{81 + 16}[/tex]
⇒ BC = [tex]\sqrt{97}[/tex]
Hence, AB = BC = √97
In the given quadrilateral, the two pair is of equal length and these sides are adjacent to each other.
Hence, it follows the property of kite.
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