Original exam question

The diagram shows quadrilateral OACB.

M is the midpoint of OA.
N is the point on BC such that BN:NC = 4 :5
OA = a
OB = b
AC = kb where k is a positive integer.
(a) Express MN in terms of k, a and b.
Give your answer in its simplest form.
(b) Is MN parallel to OB?
Give a reason for your answer.

Today we're tackling a challenging GCSE vectors question that combines several key skills: expressing vectors in terms of given quantities, working with midpoints and ratios, and determining whether vectors are parallel. This type of question appears regularly on AQA, Edexcel and OCR GCSE Higher Tier papers.

Understanding the Problem

Let's break down what we know from the question:

We have quadrilateral OACB

M is the midpoint of OA

N lies on BC with BN:NC = 4:5

We're given that OA = a, OB = b, and AC = kb (where k is a positive integer)

We need to find MN in terms of k, a, and b

We need to determine if MN is parallel to OB

flowchart TD
    O["Point O"] -->|vector a| A["Point A"]
    O -->|vector b| B["Point B"]
    A -->|vector kb| C["Point C"]
    B -->|vector?| C
    
    subgraph Midpoints and Ratios
        M["Midpoint M on OA"]
        N["Point N on BC, BN:NC = 4:5"]
    end
    
    M -->|vector MN?| N
    
    style O fill:#e1f5fe
    style A fill:#f3e5f5
    style B fill:#e8f5e8
    style C fill:#fff3e0
    style M fill:#ffebee
    style N fill:#f1f8e9

Part (a): Expressing MN in Terms of k, a and b

Step 1: Establish our approach

To find vector MN, we need a route from M to N. A reliable strategy is to use position vectors relative to a common origin (usually O). We can say:

MN = ON - OM

Where ON is the position vector of N, and OM is the position vector of M.

Step 2: Find OM (position vector of M)

Since M is the midpoint of OA, and OA = a:

OM = ¹⁄_2OA = ¹⁄_2a

This is straightforward - the midpoint divides the vector OA in half.

Step 3: Find ON (position vector of N)

This is trickier because N lies on BC with a given ratio. First, we need to understand the quadrilateral's geometry.

From the information:

OA = a

OB = b

AC = kb

We can find BC using vector addition:

BC = BA + AC (route from B to A to C)

But BA = OA - OB = a - b (from O to A minus from O to B)

So BC = (a - b) + kb = a - b + kb = a + (k - 1)b

Step 4: Apply the ratio theorem

N divides BC in the ratio 4:5 (BN:NC). This means N is ⁴⁄₉ of the way from B to C.

Using the ratio theorem:

ON = OB + (⁴⁄₉)BC

Substituting what we know:

OB = b
BC = a + (k - 1)b

So: ON = b + (⁴⁄₉)[a + (k - 1)b]

Step 5: Simplify ON

ON = b + (⁴⁄₉)a + (⁴⁄₉)(k - 1)b

Combine the b terms:
ON = (⁴⁄₉)a + [1 + (⁴⁄₉)(k - 1)]b

Simplify the coefficient of b:
1 + (⁴⁄₉)(k -
1) = 1 + (^4k⁄₉) - (⁴⁄₉) = (⁵⁄₉) + (^4k⁄₉) = (4k + ⁵⁾⁄₉

So: ON = (⁴⁄₉)a + [(4k + ⁵⁾⁄₉]b

Step 6: Calculate MN

MN = ON - OM = [(⁴⁄₉)a + ((4k + ⁵⁾⁄_9)b] - (¹⁄_2a)

Combine the a terms:
(⁴⁄₉)a - (¹⁄₂)a = (⁴⁄₉)a - (⁹⁄₁₈)a = (⁸⁄₁₈)a - (⁹⁄₁₈)a = -(¹⁄₁₈)a

So: MN = -(¹⁄₁₈)a + [(4k + ⁵⁾⁄₉]b

We can write the b term with denominator 18 for consistency:
[(4k + ⁵⁾⁄₉] = [2(4k + ⁵⁾⁄₁₈] = (8k + ¹⁰⁾⁄₁₈

Therefore: MN = -(¹⁄₁₈)a + [(8k + ¹⁰⁾⁄₁₈]b

Or factorising ¹⁄₁₈: MN = (¹⁄₁₈)[-a + (8k + 10)b]

This is our simplified expression for part (a).

Part (b): Is MN parallel to OB?

Step 1: Understanding parallel vectors

Two vectors are parallel if one is a scalar multiple of the other. OB = b, so MN is parallel to OB if MN = λb for some scalar λ.

Looking at our expression for MN:

MN = (¹⁄₁₈)[-a + (8k + 10)b]

Step 2: Analyse the components

For MN to be parallel to b, it must have no a component. That means the coefficient of a must be zero.

In our expression, the a component is -(¹⁄₁₈). This is not zero unless... wait, let's check carefully.

The full expression is: MN = -(¹⁄₁₈)a + [(8k + ¹⁰⁾⁄₁₈]b

The a component is -(¹⁄₁₈), which is a fixed number, not dependent on k.

Step 3: Draw our conclusion

Since the a component is -(¹⁄₁₈) and not zero (and cannot be made zero by choosing k, as k only affects the b component), MN always has an a component.

Therefore, MN is NOT parallel to OB because it contains an a term, whereas OB is purely in terms of b.

Key Learning Points

Position vectors are your friend: When finding vectors between points, express them relative to a common origin.

Ratio theorem: For a point P dividing AB in ratio m:n, OP = (nOA + ^mOB)⁄_(m+n)

Parallel vectors: Vectors are parallel if one is a scalar multiple of the other (no other components).

Midpoints: The midpoint M of AB gives OM = ¹⁄₂(OA + OB)

Common Mistakes to Avoid

Forgetting that vectors have direction: OA and AO are different

Misapplying the ratio theorem: ensure you use the correct fractions

Not simplifying expressions fully

Confusing position vectors with displacement vectors

Practice This Method

Try this variation: What if BN:NC was 5:4 instead? How would MN change? Would it ever be parallel to OB with different ratios?

Remember, vector questions in GCSE maths test your understanding of geometry through algebra. Practice breaking down diagrams into vector pathways, and you'll master this topic!

Need more help with vectors? Check out our other posts on vector proofs and geometric applications.

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