Question

The coordinates of the endpoints of AB and CD are A(2,

3), B(8, 1), C(5,2), and D(6,5). Which statement about the
line segments is true?
• AB 1 CD, but CD does not bisect AB.
• AB I CD, and CD bisects AB.
• AB is not I to CD, but CD bisects AB.
• AB is not I to CD, and CD does not bisect AB.​

Answer 1

Option 1: CD is a perpendicular bisector of AB

Explanation:

Let us find out the slopes of various line segments and the Distances and then we will draw the conclusions accordingly.

Formula to find slope

`m= (y_2-y_1)/(x_2-x_1)`

Formula to Find Distance between two points

`D=√((y_2-y_1)^2+(x_2-x_1)^2)`

mAB ( represents , Slope of AB )

1.
`mAC= (3-2)/(2-5)=(1)/(-3)=-(1)/(3)`

2.
`mBC=(2-1)/(5-8)=(1)/(-3)=-(1)/(3)`

3.
`mCD=(5-2)/(6-5)=(3)/(1)=3`

4.
`AC=√((3-2)^2+(2-5)^2) =√((1)^2+(-3)^2)=√(1+9)=√(10)`

5.
`BC=√((2-1)^2+(5-8)^2) =√((1)^2+(-3)^2)=√(1+9)=√(10)`

mAC = mBC , and C is common point , hence these three are collinear points making a straight line whole slope is
`-(1)/(3)`

`mAB=-(1)/(3)`

`mCD=3`

`mAB * mCD = -(1)/(3) * 3 = -1`

Hence CD ⊥ AB

Also

From Point 4 and point 5 above , we see that

AC = CB

Hence CD bisect AB at C, also CD ⊥ AB

There fore

CD is a perpendicular bisector of AB

Therefor option 1 is true