Question

The endpoints of AB are A(–8, –6) and B(4, 10). The midpoint is at C(–2, 2). What is true about the point D, the midpoint of CB? Check all that apply.

The coordinates of D are (2, 6).
D partitions AB in a 2:1 ratio.
The coordinates of D are (1, 6).
C partitions AD in a 2:1 ratio.
D partitions AB in a 3:1 ratio.

Answer 1

Its the 3rd 4th and 5th options

Answer 2

3rd, 4th and 5th options are correct choices.

Explanation:

We have been given that the endpoints of AB are A(–8, –6) and B(4, 10). The midpoint is at C(–2, 2). The point D is the midpoint of CB.

Let us find the coordinates of D using midpoint formula.

`\text{Midpoint}=((x_1+x_2)/(2),(y_1+y_2)/(2))`

Upon substituting coordinates of point C and B is midpoint formula we will get,

`\text{Midpoint of CB}=((-2+4)/(2),(2+10)/(2))`

`\text{Midpoint of CB}=((2)/(2),(12)/(2))`

`\text{Midpoint of CB}=(1,6)`

Since D is the midpoint of CB, therefore, the coordinates of point D are (1,6) and 3rd option is the correct choice.

Let us find the length of each segment using distance formula.

`\text{Distance}=√((x_2-x_1)^2+(y_2-y_1)^2)`

`\text{Length of segment AC}=√((-2--8)^2+(2--6)^2)`

`\text{Length of segment AC}=√((-2+8)^2+(2+6)^2)`

`\text{Length of segment AC}=√((6)^2+(8)^2)`

`\text{Length of segment AC}=√(36+64)`

`\text{Length of segment AC}=√(100)`

`\text{Length of segment AC}=10`

`\text{Length of segment CD}=√((1--2)^2+(6-2)^2)`

`\text{Length of segment CD}=√((1+2)^2+(4)^2)`

`\text{Length of segment CD}=√((3)^2+(4)^2)`

`\text{Length of segment CD}=√(9+16)`

`\text{Length of segment CD}=√(25)=5`

`\text{Length of segment AD}=√((1--8)^2+(6--6)^2)`

`\text{Length of segment AD}=√((1+8)^2+(6+6)^2)`

`\text{Length of segment AD}=√((9)^2+(12)^2)`

`\text{Length of segment AD}=√(81+144)`

`\text{Length of segment AD}=√(225)=15`

`\text{Length of segment DB}=√((1-4)^2+(6-10)^2)`

`\text{Length of segment DB}=√((3)^2+(-4)^2)`

`\text{Length of segment DB}=√(9+16)`

`\text{Length of segment DB}=√(25)=5`

Now let us check our 2nd, 4th and 5th options one by one.

2. D partitions AB in a 2:1 ratio.

We can represent this information using proportion as:

`(AD)/(DB)=(2)/(1)`

Upon substituting length of AD and DB we will get,

`(15)/(5)=(2)/(1)`

`(3)/(1)\\eq(2)/(1)`

Since both sides of our equation are not equal, therefore, 2nd option is not a correct choice.

4. C partitions AD in a 2:1 ratio.

We can represent this information using proportion as:

`(AC)/(CD)=(2)/(1)`

Upon substituting length of AC and CD we will get,

`(10)/(5)=(2)/(1)`

`(2)/(1)=(2)/(1)`

Since both sides of our equation are equal, therefore, 4th option is a correct choice.

5. D partitions AB in a 3:1 ratio.

We can represent this information using proportion as:

`(AD)/(DB)=(3)/(1)`

Upon substituting length of AD and DB we will get,

`(15)/(5)=(3)/(1)`

`(3)/(1)=(3)/(1)`

Since both sides of our equation are equal, therefore, 5th option is a correct choice.