GCSE · Maths · AQA · Spec 8300

Geometrical problems on coordinate axes

Can you prove a corner is exactly 90° without a protractor? On coordinate axes you can, with two numbers: how far a line goes across, and how far up.

Drag and discover

When are two lines parallel, and when are they perpendicular?

OA run 3.0. OA rise 1.0. CD run 3.0. CD rise 2.0. OB run (to the left) 1.0. OB rise 1.0. Angle AOB 117°. Classification: The two rules you are testing. Relationship: Parallel lines have the same gradient. Perpendicular sloping lines have gradients that multiply to −1.

OA run3.0OA rise1.0CD run3.0CD rise2.0OB run (to the left)1.0OB rise1.0Angle AOB117°drag D, then B

The two rules you are testing

Parallel lines have the same gradient. Perpendicular sloping lines have gradients that multiply to −1.

OA goes 3 across and 1 up, so its gradient is 1 ÷ 3 = 1/3. First drag D until CD runs alongside OA, and read CD's rise. Then drag B until angle AOB reads exactly 90°. Compare OB's run and rise with OA's: what has happened to the 3 and the 1?

Use the rules

Parallel, perpendicular or neither?

Pick a pair of lines, then choose the group it belongs in.

Still to sort

Parallel (0)

The two gradients are equal.

Where the line is: Equal gradients, not gradients that multiply to 1.

Perpendicular (0)

The two gradients multiply to −1.

Where the line is: The product has to be exactly −1. Flipping the fraction on its own is not enough, and neither is changing the sign on its own.

Neither (0)

Any other pair: the lines cross, but not at a right angle.

6 of 6 still to sort.

Find or read each pair's gradients, then decide. For a line through two points, gradient = change in y ÷ change in x, with up positive and down negative. Careful: a couple of these pairs are only nearly right.

Prove it on a shape

Is EFGH a rectangle?

E is (1, 2), F is (5, 4), G is (4, 6) and H is (0, 4). Use gradients to show that EFGH is a rectangle. [4 marks]

Your answer

0 words · your answer stays on this page and is not sent anywhere.

The mark scheme

Works out that EF and GH both have gradient 1/2, so EF is parallel to GH. (1 mark)
Works out that FG and HE both have gradient −2, so FG is parallel to HE. (1 mark)
Shows 1/2 × (−2) = −1, so EF is perpendicular to FG and angle EFG is 90°. (1 mark)
Concludes: two pairs of parallel sides and a right angle, so all four angles are 90° and EFGH is a rectangle. (1 mark)

Model answer

Gradient of EF = (4 − 2) ÷ (5 − 1) = 2/4 = 1/2. Gradient of GH = (4 − 6) ÷ (0 − 4) = −2/−4 = 1/2. These are equal, so EF is parallel to GH. Gradient of FG = (6 − 4) ÷ (4 − 5) = 2/−1 = −2. Gradient of HE = (2 − 4) ÷ (1 − 0) = −2. These are equal, so FG is parallel to HE. So EFGH is a parallelogram. Also 1/2 × (−2) = −1, so EF is perpendicular to FG and angle EFG = 90°. A parallelogram with one right angle has all four angles equal to 90°, so EFGH is a rectangle.

The gradients are only half the job. Each gradient fact has to be turned into words (these sides are parallel, this angle is 90°), and the last line has to name the shape.

Now: points in between

Where exactly is P?

A line segment runs from A to B. Point P sits on it so that AP : PB = 1 : 3.

Which is closest to what you think right now?

P is a third of the way from A to B.P is a quarter of the way from A to B.P is three quarters of the way from A to B.

How sure are you?

I'm sureI think soI'm guessing

P is a third of the way from A to B.

This is a common misconception

Really tempting, because there's a 3 right there. But 1 : 3 compares the two pieces with each other, not a piece with the whole line. AP is 1 part and PB is 3 parts, so the whole of AB is 4 equal parts. P is 1 part along out of 4, which is a quarter of the way.

P is a quarter of the way from A to B.

This matches the evidence

Yes. 1 part + 3 parts = 4 equal parts, and P is 1 of them away from A. So AP is 1/4 of AB, and PB is the other 3/4.

P is three quarters of the way from A to B.

This is a common misconception

You counted 4 parts, and that's the key step. Now check which piece is which. AP is the piece touching A, and it is the 1 part. So P is 1/4 of the way from A. Three quarters is how far P is from B.

Add the parts before you do anything else. The total number of parts is the bottom of your fraction, never the other part of the ratio.

Why it works on a slope

Reason it through

Why can you take the same fraction of the x-distance and of the y-distance, even when AB slopes?

Link 1 of 4

First link · your turn

P is a third of the way along a sloping line from A to B. Draw the slope triangle from A to B: its horizontal side is the x-distance and its vertical side is the y-distance. Now draw the smaller slope triangle from A to P. What do the two triangles have in common?

Locked — reveal the link above first

Worked example: finding P

Problem

A is (−3, 8) and B is (7, −2). P lies on AB so that AP : PB = 2 : 3. Find the coordinates of P.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Equal gradients mean parallel. Gradients that multiply to −1 mean perpendicular. And one fraction, used across and up, finds any point in between two others.

What you need to know

Gradient = rise ÷ run = change in y ÷ change in x. Going down, or going left, counts as negative.
Parallel lines have the same gradient.
Perpendicular sloping lines have gradients that multiply to −1. Flip the fraction and change the sign: 1/3 goes with −3. A horizontal line and a vertical line are perpendicular too, but a vertical line has no gradient, so the −1 rule can't be used on that pair.
To divide a line segment in a ratio, add the parts first. P is (AP's parts ÷ total parts) of the way from A to B.
Take that fraction of the x-distance and of the y-distance from A to B, then add the results to A's coordinates.

The big picture

Everything here runs on rise and run. Compare two sloping lines' gradients and you know whether they are parallel (equal), perpendicular (multiply to −1) or neither, and inside a shape that tells you which sides are parallel and which corners are right angles. (A horizontal line and a vertical line are perpendicular too, but a vertical line has no gradient, so the −1 test is only for sloping lines.) For a point partway along a segment, turn the ratio into a fraction of the whole, then use that same fraction on the x-distance and the y-distance from A.

Key points

1To show two lines are parallel, work out both gradients and show they are equal.

2To show two sloping lines are perpendicular, work out both gradients and show they multiply to −1. (A horizontal and a vertical line are perpendicular as well, but a vertical line has no gradient, so this test can't be used on them.)

3In a shape, two sloping sides meeting at a corner with gradients that multiply to −1 make a right angle there. A horizontal side meeting a vertical side makes a right angle too.

4Finish a shape proof in words: say which sides are parallel, which angle is 90°, and what the shape is.

5The ratio works on x and y separately because the slope triangle from A to P is similar to the one from A to B.

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Worked example

Problem

Line L passes through (−2, 4) and (4, 0). Find the gradient of a line that is perpendicular to L.

⚠ Watch out

Taking the fraction of the coordinates instead of the distances. If A is (4, 2), B is (12, 6) and P is a quarter of the way along, a quarter of B is (3, 1.5), not even on AB. Quarter the distances and add to A: P is (6, 3).

🧠

Memory hook

Parallel: same slope. Perpendicular (sloping lines): flip it and switch the sign. Ratios: count ALL the parts, then go across and up by the same fraction.

✓

Check yourself

C is (1, 5) and D is (9, 1). Find the coordinates of the point that is 1/4 of the way from C to D. Then find the gradient of a line perpendicular to CD.

Flashcards

(11)

How do you find the gradient of the line through two points?

Change in y ÷ change in x (rise ÷ run).

Two lines are parallel. What do you know about their gradients?

They are equal.

Two sloping lines are perpendicular. What do you know about their gradients?

They multiply to −1. (A horizontal and a vertical line are perpendicular too, but a vertical line has no gradient, so the rule can't be used there.)

A line has gradient 5. What is the gradient of a line perpendicular to it?

−1/5, because 5 × (−1/5) = −1.

Why do the gradients of perpendicular sloping lines multiply to −1?

Turning a line through 90° swaps its rise and run and makes one of them negative.

How can gradients show that a quadrilateral is a parallelogram?

Show that both pairs of opposite sides have equal gradients.

How can gradients show that an angle in a shape is a right angle?

Show that the gradients of the two sides meeting at that corner multiply to −1.

AP : PB = 2 : 5. What fraction of the way from A to B is P?

2/7. Add the parts first: 2 + 5 = 7.

Once you know what fraction of the way P is from A to B, what do you take that fraction of?

The x-distance and the y-distance from A to B.

You have taken the fraction of each distance. What is the last step?

Add the results to A's coordinates.

Why does the same fraction work for both the x-distance and the y-distance on a sloping line?

The slope triangle from A to P is similar to the one from A to B, so every side scales by the same fraction.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Parallel, perpendicular or neither?

Is EFGH a rectangle?

Where exactly is P?

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