Now that you understand the midpoint formula and how it works, practice some problems where you might need it.

1. The midsegment of a trapezoid is formed by connecting the midpoints of the non-parallel sides. Determine the coordinates of the endpoints that would form the midsegment of trapezoid ABCD.

Click on the blanks to check your answers.

E Interactive button. Assistance may be required. __________ (-2.5, 1.5) F Interactive button. Assistance may be required. __________ (3, 1.5)

2. The diameter of circles has a midpoint at the center of the circle. Use the graph below to find the center or each circle.

   

   Click on the blanks to check your answers.

   Center of Circle 1 Interactive button. Assistance may be required. __________ (1, 2)

   Center of Circle 2 Interactive button. Assistance may be required. __________ (-2, -2)

   Center of Circle 3 Interactive button. Assistance may be required. __________ (4.5, -3.5)

3. A segment is bisected into two equal parts at its midpoint and a polygon is a parallelogram if its diagonals bisect each other.

   Based on this information determine if the polygons below are parallelograms.

   

   Click on the blanks to check your answers.

   Polygon #1

   Midpoint AC Interactive button. Assistance may be required. __________ (0.5, 2)

   Midpoint BD Interactive button. Assistance may be required. __________ (0.5, 2)

   Parallelogram? Interactive button. Assistance may be required. __________ yes

   Polygon #2

   Midpoint JH Interactive button. Assistance may be required. __________ (5, 3)

   Midpoint GI Interactive button. Assistance may be required. __________ (5.5, 3)

   Parallelogram? Interactive button. Assistance may be required. __________ no
   

   Polygon #3

   Midpoint MO Interactive button. Assistance may be required. __________ (3.5, -2.5)

   Midpoint NP Interactive button. Assistance may be required. __________ (3.5, -2.5)

   Parallelogram? Interactive button. Assistance may be required. __________ yes

Now that you can find the midpoint of a segment if you know the endpoints, can you find a missing endpoint if you know the midpoint and one endpoint?

This is working the midpoint formula backwards.

If we have our formula, M = ( x sub 1 plus x sub 2 all divided by 2 x₁ + x₂ 2 , y sub 1 plus y sub 2 all divided by 2 y₁ + y₂ 2 ) and we know that one endpoint is (4, 6) and the midpoint is (-3, 2), then we substitute in the things we know.

Like this: M = ( x sub 1 plus x sub 2 all divided by 2 x₁ + x₂ 2 , y sub 1 plus y sub 2 all divided by 2 y₁ + y₂ 2 )

(-3, 2) = ( 4 plus x sub 2 all divided by 2 4 + x₂ 2 , 6 plus y sub 2 all divided by 2 6 + y₂ 2 )

What we now have is two separate problems that we can solve one at a time.

Let's start with the unknown x value of our missing endpoint x₂ .

-3 = 4 plus x sub 2 all divided by 2 4 + x₂ 2
(2)(-3) = 4 + x₂
-6 = 4 + x₂
-6 − 4 = x₂
-10 = x₂

Now we need to find our missing endpoint y₂ .

2 = 6 plus y sub 2 all divided by 2 6 + y₂ 2
(2)(2) = 6 + y₂
4 = 6 + y₂
4 − 6 = y₂
-2 = y₂

Our missing point is (-10, -2).

Here are a few problems to solve where you might need to work the midpoint formula backwards.

1. The circle below shows the radius. Find the endpoints of the diameter.

   Click on the blanks to check your answers.

   A Interactive button. Assistance may be required. __________ (5, -6) and B Interactive button. Assistance may be required. __________ (-1, 0)

2. UV is the diameter of circle W. If one endpoint of the diameter is (5, -15) and the center of the circle is (10, -2), what are the coordinates of the endpoint of the diameter?
   
   Interactive popup. Assistance may be required.

   Check Your Answer

   (15, 11)