Graphing on the number plane
Plot and read points in all four quadrants of the number plane, find midpoints and reflections, and calculate distances between points using Pythagoras' theorem.
Worked examples
Plotting a point and identifying its quadrant
Straightforward
Problem
Plot the point on the number plane and state which quadrant it lies in.
1
Identify the x-coordinate and y-coordinate.
The x-coordinate is and the y-coordinate is .
2
Start at the origin. Move horizontally according to the x-coordinate.
Since , move 3 units to the left.
3
From that position, move vertically according to the y-coordinate.
Since , move 2 units up. Mark the point .
4
Identify the quadrant by checking the signs of the coordinates.
The x-coordinate is negative and the y-coordinate is positive. This is Quadrant 2.
Answer
Quadrant 2
Finding the midpoint of two points
Moderate
Problem
Find the midpoint of the points and .
1
Write down the midpoint formula.
2
Calculate the x-coordinate of the midpoint.
3
Calculate the y-coordinate of the midpoint.
4
Write the midpoint as an ordered pair.
Answer
Finding the distance between two points
Challenging
Problem
Find the distance between the points and .
1
Find the horizontal distance between the two points.
2
Find the vertical distance between the two points.
3
Apply Pythagoras' theorem. The horizontal and vertical distances form the two shorter sides of a right-angled triangle, and the distance is the hypotenuse.
Answer
units
Practise
Q1·Straightforward
What are the coordinates of the point that is 5 units to the right and 3 units up from the origin?
Explanation
Moving 5 units right gives . Moving 3 units up gives . The coordinates are .
Q2·Straightforward
The point lies in which quadrant of the number plane?
Explanation
The x-coordinate is (negative) and the y-coordinate is (positive). A point with a negative x and positive y lies in Quadrant 2.
Q3·Straightforward
Point has coordinates . What is the y-coordinate of ?
Explanation
In the ordered pair , the first number is the x-coordinate and the second number is the y-coordinate.
Q4·Straightforward
Point is 3 units to the left of the origin and 4 units down. What are the coordinates of ?
Explanation
Moving 3 units left gives . Moving 4 units down gives . The coordinates are .
Q5·Moderate
Find the midpoint of the points and .
Explanation
Midpoint x-coordinate:
Midpoint y-coordinate:
The midpoint is .
Midpoint y-coordinate:
The midpoint is .
Q6·Moderate
Point has coordinates . What are the coordinates of the reflection of in the x-axis?
Explanation
When reflecting in the x-axis, the x-coordinate stays the same and the y-coordinate changes sign.
reflects to .
reflects to .
Q7·Moderate
Point has coordinates . What are the coordinates of the reflection of in the y-axis?
Explanation
When reflecting in the y-axis, the y-coordinate stays the same and the x-coordinate changes sign.
reflects to .
reflects to .
Q8·Moderate
Find the midpoint of the points and .
Explanation
Midpoint x-coordinate:
Midpoint y-coordinate:
The midpoint is .
Midpoint y-coordinate:
The midpoint is .
Q9·Challenging
Find the distance between the points and .
Explanation
Horizontal distance:
Vertical distance:
Apply Pythagoras' theorem:
Vertical distance:
Apply Pythagoras' theorem:
Q10·Challenging
Find the distance between the points and .
Explanation
Horizontal distance:
Vertical distance:
Apply Pythagoras' theorem:
Vertical distance:
Apply Pythagoras' theorem:
Q11·Challenging
Find the distance between the points and .
Explanation
Horizontal distance:
Vertical distance:
Apply Pythagoras' theorem:
Vertical distance:
Apply Pythagoras' theorem:
Q12·Challenging
The midpoint of segment is . Point has coordinates . Find the coordinates of point .
Explanation
Using the midpoint formula:
Point has coordinates .
Point has coordinates .