Answer:
Step-by-step explanation:
To prove a quadrilateral a parallelogram we prove,
1). Length of opposite sides are equal.
2). Slopes of the opposite sides are same.
Length of AB = ![\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}](https://tex.z-dn.net/?f=%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D)
= ![\sqrt{(2+1)^2+(-5+2)^2}](https://tex.z-dn.net/?f=%5Csqrt%7B%282%2B1%29%5E2%2B%28-5%2B2%29%5E2%7D)
= ![3\sqrt{2}](https://tex.z-dn.net/?f=3%5Csqrt%7B2%7D)
Length of BC = ![\sqrt{(2-1)^2+(-5+2)^2}](https://tex.z-dn.net/?f=%5Csqrt%7B%282-1%29%5E2%2B%28-5%2B2%29%5E2%7D)
= ![\sqrt{10}](https://tex.z-dn.net/?f=%5Csqrt%7B10%7D)
Length of CD = ![\sqrt{(1+2)^2+(-2-1)^2}](https://tex.z-dn.net/?f=%5Csqrt%7B%281%2B2%29%5E2%2B%28-2-1%29%5E2%7D)
= ![3\sqrt{2}](https://tex.z-dn.net/?f=3%5Csqrt%7B2%7D)
Length of AD = ![\sqrt{(-1+2)^2+(-2-1)^2}](https://tex.z-dn.net/?f=%5Csqrt%7B%28-1%2B2%29%5E2%2B%28-2-1%29%5E2%7D)
= ![\sqrt{10}](https://tex.z-dn.net/?f=%5Csqrt%7B10%7D)
Therefore, AB = CD and BC = AD (Opposite sides are equal in length)
Slope of AB = ![\frac{y_2-y_1}{x_2-x_1}](https://tex.z-dn.net/?f=%5Cfrac%7By_2-y_1%7D%7Bx_2-x_1%7D)
= ![\frac{-5+2}{2+1}](https://tex.z-dn.net/?f=%5Cfrac%7B-5%2B2%7D%7B2%2B1%7D)
= -1
Slope of BC = ![\frac{-5+2}{2-1}](https://tex.z-dn.net/?f=%5Cfrac%7B-5%2B2%7D%7B2-1%7D)
= -3
Slope of CD = ![\frac{1+2}{-2-1}](https://tex.z-dn.net/?f=%5Cfrac%7B1%2B2%7D%7B-2-1%7D)
= -1
Slope of AD = ![\frac{-2-1}{-1+2}](https://tex.z-dn.net/?f=%5Cfrac%7B-2-1%7D%7B-1%2B2%7D)
= -3
Slope of AB = slope of CD and slope of BC = slope AD
Therefore, AB║CD and BC║AD
Hence ABCD is a parallelogram