AQA GCSE Mathematics Paper 1 (Higher), June 2025: Question 22

Mark scheme

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Q Answer Mark Comments

Correct method for finding the eg 6 – (– 18) or 24

difference between the x or y

or

coordinates for the points E and G

(positive or negative differences) M1 3 – 9 or –6

implied by gradient = –

Correct method for finding the −18 − 6

difference between the x or y eg or –8 or –16

coordinates for the points E and F

or the points F and G M1dep or

(positive or negative differences) 9 – 3

or 2 or 4

(–10, 7) A1 SC1 (–2, 5) or (–12, 7.5) oe

Additional Guidance

SC1 (–2, 5) is for EF = EG

SC1 (–12, 7.5) is for EF = FG

Working may be seen on the diagram

(–10, …) or (…, 7) M1M1

How to answer it

AQA GCSE Mathematics • Higher Tier

Dividing a Line Segment in a Given Fraction

What this question tests

Finding horizontal ( x ) and vertical ( y ) displacements between two coordinate points.
Dealing carefully with directed negative numbers (e.g. subtracting a negative).
Applying fractional scaling to vector translations along a line segment.
Adding fractional displacements to the starting point to determine new coordinates.

Question 22 Walkthrough

Points E(−18, 9) and G(6, 3) where EF = ⅓ EG [3 marks]

📐 Step-by-Step Calculation

Step 1: Calculate the total horizontal and vertical change from E to G

Find the full shift from the starting point E(−18, 9) to the end point G(6, 3) :

Change in x = 6 − (−18) = 6 + 18 = 24
Change in y = 3 − 9 = −6

Mark Awarded: 1 Method Mark (M1) for finding either the difference in x (24) or difference in y (−6).

Step 2: Find ⅓ of each displacement

Since point F lies one-third of the way from E towards G:

x-step = ⅓ × 24 = 8
y-step = ⅓ × (−6) = −2

Mark Awarded: 1 Dependent Method Mark (M1dep) for dividing the coordinate difference by 3 (obtaining 8 or −2).

Step 3: Add the fractional shifts to the coordinates of E

Start at E(−18, 9) and add each step:

x-coordinate of F = −18 + 8 = −10
y-coordinate of F = 9 + (−2) = 7

Mark Awarded: 1 Accuracy Mark (A1) for final answer (−10, 7) .

✅ Final Answer

(−10, 7)

Partial credit: Getting one coordinate correct, e.g. (−10, ...) or (..., 7) , earns M1M1.

💡 Key Knowledge

Vector Form: Position vector of F is F = E + ⅓(G − E) .
Direction Matters: Movement is from E towards G. Since y drops from 9 to 3, the vertical step must be subtracted (or added as a negative).

🧠 Exam Technique & Examiner Tips

Check by inspection on the sketch: Point E has x = −18 and G has x = 6. Point F must lie between them, closer to E. A value of x = −10 and y = 7 fits visually.
Label your diagram: Drawing a horizontal-vertical right-angled triangle between E and G makes the differences of 24 and 6 immediately clear.

❌ Common Errors & Mark Scheme Traps

Measuring from the wrong end: Calculating ⅓ from G instead of E gives (−2, 5) . This earns only SC1 (Special Case 1 mark).
Ratio misinterpretation: Treating EF = ⅓ EG as a ratio of 1 : 3 instead of a fraction of the whole gives a factor of ¼, leading to (−12, 7.5) . This also only scores SC1.
Sign slip: Calculating 6 − 18 = −12 instead of 6 − (−18) = +24. Always use brackets when subtracting negatives!

Topics

Algebra · Geometry and measures · 3.2.2 Graphs · 3.4.1 Properties and constructions