Answer:
Distance of AB=Distance of CD
and Distance of BC=Distance of DA
Therefore the opposite sides of rectangle ABCD are congruent to each other
Step-by-step explanation:
Let ABCD be the rectangle
and the points A(4,-7), B(4,-2), C(0,-2), D(0,-7)
Now to verify that the given points can figure as a rectangle by using distance formula
Distance of AB
Let
be the points A(4,-7), B(4,-2)
![d=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28x_%7B2%7D-x_%7B1%7D%29%5E2%2B%28y_%7B2%7D-y_%7B1%7D%29%5E2%7D)
![d=\sqrt{(4-4)^2+(-2-(-7))^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%284-4%29%5E2%2B%28-2-%28-7%29%29%5E2%7D)
![d=\sqrt{(0)^2+(5)^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%280%29%5E2%2B%285%29%5E2%7D)
![d=\sqrt{(5)^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%285%29%5E2%7D)
Therefore d=5units
Distance of BC
Let
be the points B(4,-2) and C(0,-2)
![d=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28x_%7B2%7D-x_%7B1%7D%29%5E2%2B%28y_%7B2%7D-y_%7B1%7D%29%5E2%7D)
![d=\sqrt{(0-4)^2+(-2-(-2))^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%280-4%29%5E2%2B%28-2-%28-2%29%29%5E2%7D)
![d=\sqrt{(-4)^2+(0)^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28-4%29%5E2%2B%280%29%5E2%7D)
![d=\sqrt{16}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B16%7D)
Therefore d=4units
Distance of CD
Let
be the points C(0,-2) and D(0,-7)
![d=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28x_%7B2%7D-x_%7B1%7D%29%5E2%2B%28y_%7B2%7D-y_%7B1%7D%29%5E2%7D)
![d=\sqrt{(0-0)^2+(-7-(-2))^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%280-0%29%5E2%2B%28-7-%28-2%29%29%5E2%7D)
![d=\sqrt{(0)^2+(-5)^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%280%29%5E2%2B%28-5%29%5E2%7D)
![d=\sqrt{25}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B25%7D)
Therefore d=5units
Distance of DA
Let
be the points D(0,-7) and A(4,-7)
![d=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28x_%7B2%7D-x_%7B1%7D%29%5E2%2B%28y_%7B2%7D-y_%7B1%7D%29%5E2%7D)
![d=\sqrt{(4-0)^2+(-7-(-7))^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%284-0%29%5E2%2B%28-7-%28-7%29%29%5E2%7D)
![d=\sqrt{(4)^2+(0)^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%284%29%5E2%2B%280%29%5E2%7D)
![d=\sqrt{16}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B16%7D)
Therefore d=4units
Therefore Distance of AB=Distance of CD
and Distance of BC=Distance of DA
Therefore the opposite sides of rectangle ABCD are congruent to each other