200. Distance Between Points

Learning Intentions

• Apply the distance formula to two coordinate points.
• Calculate horizontal and vertical differences correctly.
• Interpret distance as a positive length.

Pre-requisite Summary

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

• Know that the horizontal coordinate is the -coordinate and the vertical coordinate is the -coordinate.

• Know that horizontal difference is found by subtracting -coordinates.

• Know that vertical difference is found by subtracting -coordinates.

• Know that distance is a length, so it is always positive.

• Know that the distance formula is based on Pythagoras’ theorem:

Worked Examples

Worked Example 1

Find the distance between the two points.

and

Worked Example 2

Find the distance between the two points.

and

Worked Example 3

For each pair of points, calculate the horizontal difference and vertical difference.

a) and

b) and

c) and

Worked Example 4

Use horizontal and vertical differences to find the distance between the two points.

and

Worked Example 5

Find the distance between the two points and interpret the answer as a positive length.

and

Worked Example 6

Find the distance between the two points. Leave your answer in exact form.

and

Worked Example 7

Find the distance between the two points. Give your answer to one decimal place.

and

Problems

Problem 1

Find the distance between the two points.

and

Problem 2

Find the distance between the two points.

and

Problem 3

For each pair of points, calculate the horizontal difference and vertical difference.

a) and

b) and

c) and

Problem 4

Use horizontal and vertical differences to find the distance between the two points.

and

Problem 5

Find the distance between the two points and interpret the answer as a positive length.

and

Problem 6

Find the distance between the two points. Leave your answer in exact form.

and

Problem 7

Find the distance between the two points. Give your answer to one decimal place.

and

Exercises

Understanding and Fluency

Exercise 1

Calculate the horizontal difference and vertical difference between each pair of points.

a) and

b) and

c) and

d) and

Exercise 2

Calculate the horizontal difference and vertical difference between each pair of points.

a) and

b) and

c) and

d) and

Exercise 3

Find the distance between each pair of points.

a) and

b) and

c) and

d) and

Exercise 4

Find the distance between each pair of points.

a) and

b) and

c) and

d) and

Exercise 5

Find the distance between each pair of points. Leave your answer in exact form.

a) and

b) and

c) and

d) and

Exercise 6

Find the distance between each pair of points. Leave your answer in exact form.

a) and

b) and

c) and

d) and

Exercise 7

Find the distance between each pair of points. Give your answer to one decimal place.

a) and

b) and

c) and

d) and

Exercise 8

Copy and complete each distance formula substitution.

a) Distance between and :

b) Distance between and :

c) Distance between and :

Exercise 9

For each pair of points, state whether the distance should be written as a positive or negative value.

a) to

b) to

c) to

d) to

Exercise 10

Find the distance between each pair of points and state whether the answer is rational or irrational.

a) and

b) and

c) and

d) and

Reasoning

Exercise 11

Explain why the distance between two points cannot be negative.

Exercise 12

A student says the horizontal difference between and is , so the horizontal distance is .

Explain the mistake.

Exercise 13

A student finds the distance between and as:

Explain the mistake and describe what should happen to the horizontal and vertical differences.

Exercise 14

Explain why the distance between and is the same as the distance between and .

Exercise 15

A student finds the distance between and as:

Explain why this can still give the correct positive distance.

Exercise 16

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

a) The distance between and is .

b) The distance between and is .

c) The distance between and is .

Problem-solving

Exercise 17

A student walks from point to point on a map.

a) Calculate the horizontal difference.

b) Calculate the vertical difference.

c) Find the straight-line distance from to .

Exercise 18

Two points on a coordinate grid are and .

a) Find the horizontal difference.

b) Find the vertical difference.

c) Find the distance between and to one decimal place.

Exercise 19

A triangle has vertices , and .

a) Find the length of .

b) Find the length of .

c) Find the length of using the distance formula.

Exercise 20

Create your own pair of coordinate points where the distance between them is exactly units.

Your response must include:

• the two coordinate points
• the horizontal difference
• the vertical difference
• a distance formula calculation showing the distance is

Potential Misunderstandings

• Students may subtract the coordinates in the wrong order and think a negative difference means a negative distance.
• Students may use the distance formula without identifying the -coordinates and -coordinates correctly.
• Students may confuse the distance formula with the midpoint formula.
• Students may calculate only the horizontal or vertical difference and call it the distance.
• Students may forget to square the horizontal and vertical differences.
• Students may add coordinates directly instead of subtracting corresponding coordinates.
• Students may interpret distance as direction rather than length.
• Students may give a negative answer for distance because one coordinate difference is negative.
• Students may forget that exact distances can involve square roots, such as , while approximate distances use decimals.

Next: 201. Finding Gradient of a Line Segment