Midpoints -find the coordinates

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) what are the coordinates of D?

Answer 1

Our formula for midpoints is as follows

Part A.

Since B is the midpoint of AC, we'll find the coordinate of B first since we have the information we need.

Point A (-9,-4), we'll call (x₁,y₁)

Point C (-1,6), we'll call (x₂,y₂)

if we plug those numbers into our midpoint formula, we have:

x_m= (-9-1)/2 = -10/2 = -5

y_m= (-4+6)/2 = 2/2 = 1

Since our midpoint of AC is B, the coordinates of B are (-5,1)

Part B.

Now we have to find the coordinates of point D (x₂,y₂)

Point E, the midpoint of BD has the coordinates (-4,-3).

We'll designate (x_m,y_m) as (-4,-3) for this part.

and (x₁,y₁) as point B, (-5,1)

now we plug these values into the midpoint formula

x_m= -4= (-5+x₂)/2 , we multiply both sides of the equation to get rid of the 2 in the denominator

-8= -5+x₂ , we isolate x further by adding 5 to both sides (gets rid of -5 on right side of equation)

-3= x₂

y_m= -3=(1+y₂)/2 ,multiply both sides by 2

-6= 1+y₂ ,subtract 1 from both sides (gets rid of 1 on right side of equation)

-7= y₂

The coordinates of point D are (-3,-7)

Answer 2

To find the coordinates of D, you must know B. This is because E is the midpoint of BD, and we know E, but we need not just the midpoint but the other end, B. We know that B is the midpoint of AC, so we use the midpoint formula:

B = ( (x₁+x₂)/2, (y₁+y₂)/2 )

where A = (x₁, y₁) = (-9,-4)

and C = (x₂, y₂) = (-1,6)

Intuitively, the midpoint formula finds the middle or average point between the two endpoints. That's why you sum them and divide by 2 - that's how you take the average of two items.

Now, without giving away the full answer, you can find B using the above equation and points. We're not done, though. You are asked to find the coordinates of D. We know that E is the midpoint of BD, and we now know what E and B are (E is given and B was just calculated). How do you think we can find the endpoint D given those two other points on the same line segment?