Question

The following set of coordinates represents which figure? (7, 10), (4, 7), (6, 5), (9, 8) Parallelogram Rectangle Rhombus Square

Answer 1

The figure is a rectangle

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)}`

- The rule of the slope is
`m=(y_(2)-y_(1))/(x_(2)-x_(1))`

- Remember:

* 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)}=√(18)`

∵
`m1=(7-10)/(4-7)=(-3)/(-3)=1`

# points (4 , 7) and (6 , 5)

∵
`d2=\sqrt{(6-4)^(2)+(5-7)^(2)}=√(8)`

∵
`m2=(5-7)/(6-4)=(-2)/(2)=-1`

# points (6 , 5) and (9 , 8)

∵
`d3=\sqrt{(9-6)^(2)+(8-5)^(2)}=√(18)`

∵
`m3=(8-5)/(9-6)=(3)/(3)=1`

# points (9 , 8) and (7 , 10)

∵
`d4=\sqrt{(7-9)^(2)+(10-8)^(2)}=√(8)`

∵
`m4=(10-8)/(7-9)=(2)/(-2)=-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