Find the midpoints for the diagram below and then draw the lines of reflection.

### Watch This

First watch this video to learn about the midpoint formula.

CK-12 Foundation Chapter10TheMidpointFormulaA

Then watch this video to see some examples.

CK-12 Foundation Chapter10TheMidpointFormulaB

### Guidance

The midpoint of a line segment is the point exactly in the middle of the two endpoints. In order to calculate the coordinates of the midpoint, find the average of the two endpoints:

Sometimes midpoints can help you to find lines of reflection (lines of symmetry) in shapes. Look at the equilateral triangle in the diagram below.

In an equilateral triangle there are three lines of symmetry. The lines of symmetry connect each vertex to the midpoint on the opposite side.

Keep in mind that not all midpoints will create lines of symmetry!

#### Example A

In the diagram below,

**Solution:**

#### Example B

Find the coordinates of point

**Solution:** Look at the midpoint formula:

For this problem, if you let point

Next you need to separate the

Now multiply each of the equations by 2 in order to get rid of the fraction.

Now you can solve for

Therefore the point

#### Example C

Find the midpoints for the diagram below in order to draw the lines of reflection (or the line of symmetry).

**Solution:**

As seen in the graph above, a square has two lines of symmetry drawn from the mid-points of the opposite sides. A square actually has two more lines of symmetry that are the diagonals of the square.

#### Concept Problem Revisited

Find the midpoints for the diagram below in order to draw the lines of reflection.

As seen in the graph above, a rectangle has two lines of symmetry.

### Guided Practice

1. In the diagram below,

2. Find the coordinates of point

3. A diameter is drawn in the circle as shown in the diagram below. What are the coordinates for the center of the circle,

**Answers:**

1.

2. Let point

Next you need to separate the

Now multiply each of the equations by 2 in order to get rid of the fraction.

Now you can solve for

Therefore the point

3.

### Explore More

Find the mid-point for each line below given the endpoints:

- Line
AB givenA(5,7) andB(3,9) . - Line
BC givenB(3,8) andC(5,2) . - Line
CD givenC(4,6) andD(3,5) . - Line
DE givenD(9,11) andE(2,2) . - Line
EF givenE(1,1) andF(8,7) . - Line
FG givenF(1,8) andG(1,4) .

For the following lines, one endpoint is given and then the mid-point. Find the other endpoint.

- Line
AB givenA(3,−5) andMAB(7,7) . - Line
BC givenB(2,4) andMBC(4,9) . - Line
CD givenC(−2,6) andMCD(1,1) . - Line
DE givenD(2,9) andMDE(8,2) . - Line
EF givenE(−6,−5) andMEF(−2,6) .

For each of the diagrams below, find the midpoints.