Answer:
The figure is a rectangle
Step-by-step explanation:
* Lets explain how to solve the problem
- To prove the following set of coordinates represents which figure
lets find the distance between each two points and the slopes of
the lines joining these points
- The rule of the distance between two point is
![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%5E%7B2%7D%2B%28y_%7B2%7D-y_%7B1%7D%29%5E%7B2%7D%7D)
- The rule of the slope is ![m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7By_%7B2%7D-y_%7B1%7D%7D%7Bx_%7B2%7D-x_%7B1%7D%7D)
- <em>Remember:</em>
* Parallel lines have same slopes
* The product of the slopes of the perpendicular lines is -1
# points (7 , 10) and (4 , 7)
∵ ![d1=\sqrt{(4-7)^{2}+(7-10)^{2}}=\sqrt{18}](https://tex.z-dn.net/?f=d1%3D%5Csqrt%7B%284-7%29%5E%7B2%7D%2B%287-10%29%5E%7B2%7D%7D%3D%5Csqrt%7B18%7D)
∵ ![m1=\frac{7-10}{4-7}=\frac{-3}{-3}=1](https://tex.z-dn.net/?f=m1%3D%5Cfrac%7B7-10%7D%7B4-7%7D%3D%5Cfrac%7B-3%7D%7B-3%7D%3D1)
# points (4 , 7) and (6 , 5)
∵ ![d2=\sqrt{(6-4)^{2}+(5-7)^{2}}=\sqrt{8}](https://tex.z-dn.net/?f=d2%3D%5Csqrt%7B%286-4%29%5E%7B2%7D%2B%285-7%29%5E%7B2%7D%7D%3D%5Csqrt%7B8%7D)
∵ ![m2=\frac{5-7}{6-4}=\frac{-2}{2}=-1](https://tex.z-dn.net/?f=m2%3D%5Cfrac%7B5-7%7D%7B6-4%7D%3D%5Cfrac%7B-2%7D%7B2%7D%3D-1)
# points (6 , 5) and (9 , 8)
∵ ![d3=\sqrt{(9-6)^{2}+(8-5)^{2}}=\sqrt{18}](https://tex.z-dn.net/?f=d3%3D%5Csqrt%7B%289-6%29%5E%7B2%7D%2B%288-5%29%5E%7B2%7D%7D%3D%5Csqrt%7B18%7D)
∵ ![m3=\frac{8-5}{9-6}=\frac{3}{3}=1](https://tex.z-dn.net/?f=m3%3D%5Cfrac%7B8-5%7D%7B9-6%7D%3D%5Cfrac%7B3%7D%7B3%7D%3D1)
# points (9 , 8) and (7 , 10)
∵ ![d4=\sqrt{(7-9)^{2}+(10-8)^{2}}=\sqrt{8}](https://tex.z-dn.net/?f=d4%3D%5Csqrt%7B%287-9%29%5E%7B2%7D%2B%2810-8%29%5E%7B2%7D%7D%3D%5Csqrt%7B8%7D)
∵ ![m4=\frac{10-8}{7-9}=\frac{2}{-2}=-1](https://tex.z-dn.net/?f=m4%3D%5Cfrac%7B10-8%7D%7B7-9%7D%3D%5Cfrac%7B2%7D%7B-2%7D%3D-1)
∵ d1 = d3 = √18 and d2 = d4 = √8
∴ Each two opposite sides are equal
∵ m1 = m3 = 1 and m2 = m4 = -1
∴ Each two opposite sides are parallel
∵ m1 × m2 = 1 × -1 = -1
∵ m2 × m3 = 1 × -1 = -1
∵ m3 × m4 = 1 × -1 = -1
∵ m4 × m1 = 1 × -1 = -1
∴ Each two adjacent sides are perpendicular
- The set of coordinates represents a figure has these properties:
1. Each two opposite sides are equal
2. Each two opposite sides are parallel
3. Each two adjacent sides are perpendicular
∴ The figure is a rectangle