Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 9: find the gradient of the line through (1, 2) and (3, 8).
Answer
3. The rise is 8 − 2 = 6 and the run is 3 − 1 = 2, and 6 ÷ 2 = 3. - From Year 9: the point (−2, 5) is reflected in the y-axis. Where is its image?
Answer
(2, 5). The x-coordinate changes sign and y stays. - From Year 9: write the equation of the line with gradient 2 that crosses the y-axis at −3.
Answer
y = 2x − 3.
Words used in this lesson
- quarter turn
- a rotation of 90°. For example: from facing north to facing east is a clockwise quarter turn.
- anticlockwise
- the direction opposite to the hands of a clock. For example: the direction positive angles are measured.
- perpendicular
- meeting at a right angle. For example: the two sides of a set square at its corner.
- negative reciprocal
- flip the fraction and change its sign. For example: −7 ÷ 2 becomes 2 ÷ 7.
- perpendicular bisector
- the line through the midpoint of a segment at a right angle to it. For example: the fold line when two points are folded onto each other.
The map on the table
A surveyor in Cusco pins a paper map to a table at one corner, which is the origin, and turns the map through a quarter turn. A landmark that was 5 across and 2 up from the pin is somewhere new. The line from the pin to the landmark has turned too, and it has turned through exactly a right angle.
Predict first: where does the landmark go, and what do you think happens to the gradient of the line from the pin to it? Use the figure to test your idea for other points.
P (5, 2) turned through 0 quarter turns about O goes to (5, 2). The gradient of OP is 2/5 and the gradient of OP′ is 2/5.
Show me the steps
- Set P to (4, 3) and one quarter turn: the image is (−3, 4). The gradients are 3/4 and −4/3, and their product is −1.
- Keep P at (4, 3) and set two quarter turns: the image is (−4, −3), on the same line through the origin. Both gradients are 3/4.
- Set three quarter turns: the image is (3, −4), which is a quarter turn clockwise. The gradients are again 3/4 and −4/3.
- Set P to (5, 0) and one quarter turn: the image is (0, 5). One line is horizontal and the other vertical, so the product rule does not apply, but they are still perpendicular.
Rotation about the origin. A quarter turn anticlockwise sends (a, b) to (−b, a). A quarter turn clockwise sends (a, b) to (b, −a). A half turn sends (a, b) to (−a, −b).
Perpendicular gradients. The line from the origin to (a, b) has gradient b ÷ a, and after a quarter turn it has gradient a ÷ (−b). The product is −1. So lines with gradients m and −1 ÷ m (m not zero) are perpendicular, and the product of two perpendicular gradients is −1.
Equations. A line through (x₁, y₁) with gradient m is y − y₁ = m(x − x₁). Move every term to one side to get the form ax + by + c = 0.
Another centre. To turn about a point C, subtract C from the point, turn about the origin, then add C back.
The coordinate rule is for the origin only. To turn (7, 3) a quarter turn anticlockwise about (3, −2), first subtract the centre: (4, 5). Turn it: (−5, 4). Add the centre back: (−2, 2). Turning the coordinates (7, 3) directly would give (−3, 7), which is wrong.
Worked examples
1. Rotate (4, −7) through 90° clockwise about the origin.
- A clockwise quarter turn sends (a, b) to (b, −a).
- Here a = 4 and b = −7, so the image is (−7, −4).
2. Find the equation of the line through (6, −1) perpendicular to 3x + 2y = 5, in the form ax + by + c = 0.
- 3x + 2y = 5 is y = −(3/2)x + 5/2, with gradient −3/2.
- The perpendicular gradient is −1 ÷ (−3/2) = 2/3.
- y + 1 = (2/3)(x − 6) gives 3y + 3 = 2x − 12, so 2x − 3y − 15 = 0.
3. Rotate (3, 5) through 90° anticlockwise about (−1, 2).
- Subtract the centre: (3 − (−1), 5 − 2) = (4, 3).
- Turn: (−3, 4). Add the centre back: (−3 − 1, 4 + 2) = (−4, 6).
Your go. The first step is given. Find the equation of the line through (6, 1) perpendicular to y = 3x − 4, in the form ax + by + c = 0 with a = 1.
- The gradient of y = 3x − 4 is 3, so the perpendicular gradient is −1/3.
- Now write y − 1 = −(1/3)(x − 6) and rearrange.
Check the answer
3(y − 1) = −(x − 6), so 3y − 3 = −x + 6, which gives x + 3y − 9 = 0. Check: at (6, 1), 6 + 3 − 9 = 0.
Check before you go on
1. The point (8, −3) is rotated 90° anticlockwise about the origin. Where is its image?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. The point (10, 2) is rotated 90° clockwise about the origin. Find the y-coordinate of its image.
3. A line has equation y = −(4/5)x + 3. Find the gradient of a line perpendicular to it, as a decimal.
Level 2 · Building
4. A line has equation 2x + 5y − 20 = 0. Find the gradient of a line perpendicular to it, as a decimal.
5. A line passes through (2, 5) and is perpendicular to y = (2/3)x + 1. Find where it crosses the y-axis.
6. The line x + 2y + c = 0 is perpendicular to 2x − y + 3 = 0 and passes through (4, 1). Find c.
Level 3 · Stretch
7. The point (9, 4) is rotated 90° anticlockwise about the point (2, 1). Find the x-coordinate of its image.
8. The points A (2, 5) and B (8, 1) are given. The perpendicular bisector of AB crosses the y-axis at a point. Find its y-coordinate, to one decimal place.
Something to think about
A half turn about the origin sends the line y = 2x + 3 to another line. Which line, and how is it related to the first? What does a half turn about the point (1, 5), which is on the line, do to it?
One way to think about it
A half turn sends (x, y) to (−x, −y), so a point on the image satisfies −y = 2(−x) + 3, which is y = 2x − 3: a parallel line, with the same gradient and the intercept changed in sign. A half turn turns every line through 180°, so a gradient never changes. About (1, 5), each point of the line goes to the point the same distance from (1, 5) on the other side, which is on the same line, so the line is sent to itself.
Exit ticket
9. Which of these lines is perpendicular to 3x − 4y = 12?