Answer:
3rd, 4th and 5th options are correct choices.
Step-by-step 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}=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})](https://tex.z-dn.net/?f=%5Ctext%7BMidpoint%7D%3D%28%5Cfrac%7Bx_1%2Bx_2%7D%7B2%7D%2C%5Cfrac%7By_1%2By_2%7D%7B2%7D%29)
Upon substituting coordinates of point C and B is midpoint formula we will get,
![\text{Midpoint of CB}=(\frac{-2+4}{2},\frac{2+10}{2})](https://tex.z-dn.net/?f=%5Ctext%7BMidpoint%20of%20CB%7D%3D%28%5Cfrac%7B-2%2B4%7D%7B2%7D%2C%5Cfrac%7B2%2B10%7D%7B2%7D%29)
![\text{Midpoint of CB}=(\frac{2}{2},\frac{12}{2})](https://tex.z-dn.net/?f=%5Ctext%7BMidpoint%20of%20CB%7D%3D%28%5Cfrac%7B2%7D%7B2%7D%2C%5Cfrac%7B12%7D%7B2%7D%29)
![\text{Midpoint of CB}=(1,6)](https://tex.z-dn.net/?f=%5Ctext%7BMidpoint%20of%20CB%7D%3D%281%2C6%29)
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}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}](https://tex.z-dn.net/?f=%5Ctext%7BDistance%7D%3D%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D)
![\text{Length of segment AC}=\sqrt{(-2--8)^2+(2--6)^2}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20AC%7D%3D%5Csqrt%7B%28-2--8%29%5E2%2B%282--6%29%5E2%7D)
![\text{Length of segment AC}=\sqrt{(-2+8)^2+(2+6)^2}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20AC%7D%3D%5Csqrt%7B%28-2%2B8%29%5E2%2B%282%2B6%29%5E2%7D)
![\text{Length of segment AC}=\sqrt{(6)^2+(8)^2}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20AC%7D%3D%5Csqrt%7B%286%29%5E2%2B%288%29%5E2%7D)
![\text{Length of segment AC}=\sqrt{36+64}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20AC%7D%3D%5Csqrt%7B36%2B64%7D)
![\text{Length of segment AC}=\sqrt{100}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20AC%7D%3D%5Csqrt%7B100%7D)
![\text{Length of segment AC}=10](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20AC%7D%3D10)
![\text{Length of segment CD}=\sqrt{(1--2)^2+(6-2)^2}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20CD%7D%3D%5Csqrt%7B%281--2%29%5E2%2B%286-2%29%5E2%7D)
![\text{Length of segment CD}=\sqrt{(1+2)^2+(4)^2}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20CD%7D%3D%5Csqrt%7B%281%2B2%29%5E2%2B%284%29%5E2%7D)
![\text{Length of segment CD}=\sqrt{(3)^2+(4)^2}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20CD%7D%3D%5Csqrt%7B%283%29%5E2%2B%284%29%5E2%7D)
![\text{Length of segment CD}=\sqrt{9+16}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20CD%7D%3D%5Csqrt%7B9%2B16%7D)
![\text{Length of segment CD}=\sqrt{25}=5](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20CD%7D%3D%5Csqrt%7B25%7D%3D5)
![\text{Length of segment AD}=\sqrt{(1--8)^2+(6--6)^2}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20AD%7D%3D%5Csqrt%7B%281--8%29%5E2%2B%286--6%29%5E2%7D)
![\text{Length of segment AD}=\sqrt{(1+8)^2+(6+6)^2}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20AD%7D%3D%5Csqrt%7B%281%2B8%29%5E2%2B%286%2B6%29%5E2%7D)
![\text{Length of segment AD}=\sqrt{(9)^2+(12)^2}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20AD%7D%3D%5Csqrt%7B%289%29%5E2%2B%2812%29%5E2%7D)
![\text{Length of segment AD}=\sqrt{81+144}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20AD%7D%3D%5Csqrt%7B81%2B144%7D)
![\text{Length of segment AD}=\sqrt{225}=15](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20AD%7D%3D%5Csqrt%7B225%7D%3D15)
![\text{Length of segment DB}=\sqrt{(1-4)^2+(6-10)^2}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20DB%7D%3D%5Csqrt%7B%281-4%29%5E2%2B%286-10%29%5E2%7D)
![\text{Length of segment DB}=\sqrt{(3)^2+(-4)^2}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20DB%7D%3D%5Csqrt%7B%283%29%5E2%2B%28-4%29%5E2%7D)
![\text{Length of segment DB}=\sqrt{9+16}](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20DB%7D%3D%5Csqrt%7B9%2B16%7D)
![\text{Length of segment DB}=\sqrt{25}=5](https://tex.z-dn.net/?f=%5Ctext%7BLength%20of%20segment%20DB%7D%3D%5Csqrt%7B25%7D%3D5)
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:
![\frac{AD}{DB}=\frac{2}{1}](https://tex.z-dn.net/?f=%5Cfrac%7BAD%7D%7BDB%7D%3D%5Cfrac%7B2%7D%7B1%7D)
Upon substituting length of AD and DB we will get,
![\frac{15}{5}=\frac{2}{1}](https://tex.z-dn.net/?f=%5Cfrac%7B15%7D%7B5%7D%3D%5Cfrac%7B2%7D%7B1%7D)
![\frac{3}{1}\neq\frac{2}{1}](https://tex.z-dn.net/?f=%5Cfrac%7B3%7D%7B1%7D%5Cneq%5Cfrac%7B2%7D%7B1%7D)
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:
![\frac{AC}{CD}=\frac{2}{1}](https://tex.z-dn.net/?f=%5Cfrac%7BAC%7D%7BCD%7D%3D%5Cfrac%7B2%7D%7B1%7D)
Upon substituting length of AC and CD we will get,
![\frac{10}{5}=\frac{2}{1}](https://tex.z-dn.net/?f=%5Cfrac%7B10%7D%7B5%7D%3D%5Cfrac%7B2%7D%7B1%7D)
![\frac{2}{1}=\frac{2}{1}](https://tex.z-dn.net/?f=%5Cfrac%7B2%7D%7B1%7D%3D%5Cfrac%7B2%7D%7B1%7D)
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:
![\frac{AD}{DB}=\frac{3}{1}](https://tex.z-dn.net/?f=%5Cfrac%7BAD%7D%7BDB%7D%3D%5Cfrac%7B3%7D%7B1%7D)
Upon substituting length of AD and DB we will get,
![\frac{15}{5}=\frac{3}{1}](https://tex.z-dn.net/?f=%5Cfrac%7B15%7D%7B5%7D%3D%5Cfrac%7B3%7D%7B1%7D)
![\frac{3}{1}=\frac{3}{1}](https://tex.z-dn.net/?f=%5Cfrac%7B3%7D%7B1%7D%3D%5Cfrac%7B3%7D%7B1%7D)
Since both sides of our equation are equal, therefore, 5th option is a correct choice.