202. Finding Midpoints

Learning Intentions

• Apply the midpoint formula to two coordinate points.
• Calculate average x- and y-values accurately.
• Interpret the midpoint as the centre of a line segment.

Pre-requisite Summary

• Know that an ordered pair is written in the form .

• Know that the first coordinate gives the horizontal position and the second coordinate gives the vertical position.

• Know that the average of two numbers is found by adding them and dividing by .

• Know that the midpoint formula is:

• Know that the midpoint is halfway between two endpoints.

• Know that the midpoint has equal distance from each endpoint of the line segment.

Worked Examples

Worked Example 1

Find the midpoint of the line segment joining the two points.

and

Worked Example 2

Find the midpoint of the line segment joining the two points.

and

Worked Example 3

For each pair of points, calculate the average -value and average -value.

a) and

b) and

c) and

Worked Example 4

Find the midpoint of the line segment joining the two points.

and

Worked Example 5

Find the midpoint and interpret it as the centre of the line segment.

and

Worked Example 6

A line segment has endpoints and .

a) Find the midpoint.

b) Explain why the midpoint is halfway between the endpoints.

Worked Example 7

A line segment has endpoints and .

a) Find the midpoint.

b) Explain why the midpoint is the centre of a vertical line segment.

Problems

Problem 1

Find the midpoint of the line segment joining the two points.

and

Problem 2

Find the midpoint of the line segment joining the two points.

and

Problem 3

For each pair of points, calculate the average -value and average -value.

a) and

b) and

c) and

Problem 4

Find the midpoint of the line segment joining the two points.

and

Problem 5

Find the midpoint and interpret it as the centre of the line segment.

and

Problem 6

A line segment has endpoints and .

a) Find the midpoint.

b) Explain why the midpoint is halfway between the endpoints.

Problem 7

A line segment has endpoints and .

a) Find the midpoint.

b) Explain why the midpoint is the centre of a vertical line segment.

Exercises

Understanding and Fluency

Exercise 1

Find the midpoint of each line segment.

a) and

b) and

c) and

d) and

Exercise 2

Find the midpoint of each line segment.

a) and

b) and

c) and

d) and

Exercise 3

Find the midpoint of each line segment.

a) and

b) and

c) and

d) and

Exercise 4

Calculate the average -value and average -value for each pair of points.

a) and

b) and

c) and

d) and

Exercise 5

Copy and complete each midpoint calculation.

a) For and :

b) For and :

c) For and :

Exercise 6

Find the midpoint of each horizontal line segment.

a) and

b) and

c) and

d) and

Exercise 7

Find the midpoint of each vertical line segment.

a) and

b) and

c) and

d) and

Exercise 8

Find the midpoint of each line segment.

a) and

b) and

c) and

d) and

Exercise 9

For each pair of endpoints, state whether the midpoint has integer coordinates.

a) and

b) and

c) and

d) and

Exercise 10

Find the midpoint and state whether it is the centre of a horizontal, vertical or slanted line segment.

a) and

b) and

c) and

d) and

Reasoning

Exercise 11

Explain why the midpoint formula uses the average of the two -coordinates and the average of the two -coordinates.

Exercise 12

A student says the midpoint of and is because they added the coordinates.

Explain the mistake and give the correct midpoint.

Exercise 13

A student finds the midpoint of and as:

Explain the sign error and give the correct midpoint.

Exercise 14

Explain why the midpoint of and must have -coordinate .

Exercise 15

Explain why the midpoint of and must have -coordinate .

Exercise 16

Decide whether each statement is true or false. Justify your answer.

a) The midpoint of and is .

b) The midpoint of and is .

c) The midpoint of and is .

Problem-solving

Exercise 17

A map shows two towns at and .

a) Find the midpoint between the towns.

b) Interpret the midpoint in context.

c) Explain why the midpoint is not found by measuring from the origin.

Exercise 18

A line segment has endpoints and .

a) Find the midpoint.

b) Calculate the horizontal change from to the midpoint.

c) Calculate the horizontal change from the midpoint to .

Exercise 19

A line segment has endpoints and .

a) Find the midpoint.

b) Explain why the midpoint is directly above or below both endpoints.

c) State the vertical distance from the midpoint to each endpoint.

Exercise 20

Create your own pair of coordinate points with midpoint .

Your response must include:

• the two endpoints
• the midpoint calculation
• a sentence explaining why is the centre of the line segment

Potential Misunderstandings

• Students may add the coordinates but forget to divide by .
• Students may average the -coordinates correctly but forget to average the -coordinates.
• Students may reverse the order of coordinates and treat as .
• Students may make sign errors when adding negative coordinates, such as .
• Students may think the midpoint must always have whole-number coordinates.
• Students may incorrectly use the distance formula instead of the midpoint formula.
• Students may interpret the midpoint as the point closest to the origin rather than the centre of the line segment.
• Students may not recognise that horizontal line segments keep the same -coordinate at the midpoint.
• Students may not recognise that vertical line segments keep the same -coordinate at the midpoint.

Next: 203r. Recognise Linear Change