Answer:
Rectangle
Step-by-step explanation:
Kite is a quadrilateral in which exactly two pair of adjacent sides are equal.
Rectangle is a quadrilateral in which opposite sides are equal.
Square and rhombus are a quadrilaterals in which all sides are equal.
Given points: ![(-5,2), (-3,4), (1,0), (-1,-2)](https://tex.z-dn.net/?f=%28-5%2C2%29%2C%20%28-3%2C4%29%2C%20%281%2C0%29%2C%20%28-1%2C-2%29)
We need to find whether these points represent coordinates of kite , rectangle , rhombus or square .
Let the points be ![A(-5,2), B(-3,4), C(1,0), D(-1,-2)](https://tex.z-dn.net/?f=A%28-5%2C2%29%2C%20B%28-3%2C4%29%2C%20C%281%2C0%29%2C%20D%28-1%2C-2%29)
We know that distance between points
is given by ![\sqrt{\left ( x_2-x_1 \right )^2+\left ( y_2-y_1 \right )^2}](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cleft%20%28%20x_2-x_1%20%5Cright%20%29%5E2%2B%5Cleft%20%28%20y_2-y_1%20%5Cright%20%29%5E2%7D)
![AB=\sqrt{\left (-3+5 \right )^2+\left (4-2 \right )^2}=\sqrt{4+4}=\sqrt{8}\\BC=\sqrt{\left ( 1+3 \right )^2+\left ( 0-4 \right )^2}=\sqrt{16+16}=\sqrt{32}\\CD=\sqrt{\left ( -1-1 \right )^2+\left ( -2-0 \right )^2}=\sqrt{4+4}=\sqrt{8}\\AD=\sqrt{\left ( -1+5 \right )^2+\left ( -2-2 \right )^2}=\sqrt{16+16}=\sqrt{32}](https://tex.z-dn.net/?f=AB%3D%5Csqrt%7B%5Cleft%20%28-3%2B5%20%5Cright%20%29%5E2%2B%5Cleft%20%284-2%20%5Cright%20%29%5E2%7D%3D%5Csqrt%7B4%2B4%7D%3D%5Csqrt%7B8%7D%5C%5CBC%3D%5Csqrt%7B%5Cleft%20%28%201%2B3%20%5Cright%20%29%5E2%2B%5Cleft%20%28%200-4%20%5Cright%20%29%5E2%7D%3D%5Csqrt%7B16%2B16%7D%3D%5Csqrt%7B32%7D%5C%5CCD%3D%5Csqrt%7B%5Cleft%20%28%20-1-1%20%5Cright%20%29%5E2%2B%5Cleft%20%28%20-2-0%20%5Cright%20%29%5E2%7D%3D%5Csqrt%7B4%2B4%7D%3D%5Csqrt%7B8%7D%5C%5CAD%3D%5Csqrt%7B%5Cleft%20%28%20-1%2B5%20%5Cright%20%29%5E2%2B%5Cleft%20%28%20-2-2%20%5Cright%20%29%5E2%7D%3D%5Csqrt%7B16%2B16%7D%3D%5Csqrt%7B32%7D)
Here, AB = CD and AD = BC, so the given points are coordinates of the rectangle .