Answer:
- AB = AD = √53
- BC = DC = √265
- ABCD is a kite
Explanation:
Given quadrilateral ABCD with vertices A(4, 5), B(-3, 3), C(-6, -13), and D(6, -2), you want the lengths of AB, AD, BC, DC and a determination whether the figure is a kite.
Side lengths
The length AB can be described as |A -B|, where this value is computed as ...
|A -B| = √((Ax -Bx)² +(Ay -By)²) . . . . . . distance from A to B
The attached calculator display shows the distances between adjacent pairs of points are ...
(AB, BC, CD, DA) = (√53, √265, √265, √53)
This lets us fill in the form as follows:
- AB = √53
- AD = √53
- BC = √265
- DC = √265
Kite
These lengths tell us that adjacent pairs of sides are congruent, which makes the figure meet the definition of a kite.
ABCD is a kite.
__
Additional comment
It is useful to know that |a +bi| = √(a² +b²). Treating the coordinates as complex numbers is a shortcut to having the calculator find the distances. We only have to enter each vertex coordinate once, and let the list handling features of the calculator perform differences of adjacent vertices and use those to find the lengths. (Second attachment)
Doing this the long way, we have, for example, ...
|A -B| = |(4, 5) -(-3, 3)| = |(4+3, 5-3)| = |(7, 2)| = √(7² +2²) = √(49 +4) = √53
<95141404393>