Question

B is the midpoint of AC and E' is the midpoint of BD. If A(-9,-4), C(-1, 6), and E(-4,-3), find the coordinates of D.

Answer 1

The coordinates for D are (-4, -7)

First we must locate point B as it is vital to finding the midpoint of BD. To do this, we take the average of the endpoints AC since B is its midpoint.

x values = -9 + 1 = -8
Then divide by 2 for the average -8/2 = -4

y values = -4 + 6 = 2
Then divide by 2 for the average 2/2 = 1

Therefore B must be (-4, 1)

Now we know the values of E must be the average of B and D. So we can write equations for each coordinate since we know they are averages.

x - values = (Bx + Dx)/2 = Ex
(-4 + Dx)/2 = -4 ---> multiply both sides by 2
-4 + Dx = -8 ---> add -4 to both sides
Dx = -4

y - values = (By + Dy)/2 = Ey
(1 + Dy)/2 = -3 ---> multiply both sides by 2
1 + Dy = -6 ---> subtract 1 from both side
Dy = -7

So the coordinates for D must be (-4, -7)

Answer 2

The coordinates of D are:

(-3,-7)

Explanation:

We know that if a point B(c,d) is located in middle of two points i.e. A(a,b) and C(a',b').

Then the coordinates of B is given by:

`c=(a+a')/(2)\ and\ d=(b+b')/(2)`

It is given that:

B is the midpoint of AC.

A(-9,-4), C(-1, 6).

Hence, the coordinates of B(x,y) is given by:

`x=(-9-1)/(2)\ and\ y=(-4+6)/(2)\\\\x=(-10)/(2)\ and\ y=(2)/(2)\\\\x=-5\ and\ y=1`

i.e. the coordinates of B are: (-5,1).

E is the midpoint of BD.

Let the coordinates of D be (w,z)

Also, E is located at E(-4,-3).

i.e.

`-4=(-5+w)/(2)\ \ and\ \ -3=(1+z)/(2)\\\\-4* 2=-5+w\ \ and\ \ -3* 2=1+z\\\\-8=-5+w\ \ and\ \ -6=1+z\\\\w=-8+5\ \ and\ \ z=-6-1\\\\w=-3\ \ and\ \ z=-7`

Hence, the coordinates of D are: (-3,-7)