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
[tex]d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}[/tex]
- The rule of the slope is [tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
- Remember:
* Parallel lines have same slopes
* The product of the slopes of the perpendicular lines is -1
# points (7 , 10) and (4 , 7)
∵ [tex]d1=\sqrt{(4-7)^{2}+(7-10)^{2}}=\sqrt{18}[/tex]
∵ [tex]m1=\frac{7-10}{4-7}=\frac{-3}{-3}=1[/tex]
# points (4 , 7) and (6 , 5)
∵ [tex]d2=\sqrt{(6-4)^{2}+(5-7)^{2}}=\sqrt{8}[/tex]
∵ [tex]m2=\frac{5-7}{6-4}=\frac{-2}{2}=-1[/tex]
# points (6 , 5) and (9 , 8)
∵ [tex]d3=\sqrt{(9-6)^{2}+(8-5)^{2}}=\sqrt{18}[/tex]
∵ [tex]m3=\frac{8-5}{9-6}=\frac{3}{3}=1[/tex]
# points (9 , 8) and (7 , 10)
∵ [tex]d4=\sqrt{(7-9)^{2}+(10-8)^{2}}=\sqrt{8}[/tex]
∵ [tex]m4=\frac{10-8}{7-9}=\frac{2}{-2}=-1[/tex]
∵ 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
