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OAB is a triangle - Edexcel - GCSE Maths - Question 22 - 2018 - Paper 1

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Question 22

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OAB is a triangle. OPM and APN are straight lines. M is the midpoint of AB. $$ \vec{OA} = a \ \vec{OB} = b $$ Work out the ratio ON : NB.

Worked Solution & Example Answer:OAB is a triangle - Edexcel - GCSE Maths - Question 22 - 2018 - Paper 1

Step 1

Process to find \( \vec{OM} \)

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Answer

Since M is the midpoint of AB, we can express ( \vec{OM} ) as:

OM⃗=12(OA⃗+OB⃗)=12(a+b)\vec{OM} = \frac{1}{2}(\vec{OA} + \vec{OB}) = \frac{1}{2}(a + b)

Step 2

Process to find \( \vec{AP} \)

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Answer

Using the coordinates of point P on line ON, we can write:

OP⃗=x⋅OA⃗+y⋅OB⃗\vec{OP} = x \cdot \vec{OA} + y \cdot \vec{OB}

where ( x + y = 1 ) and for ( OP:PM = 3:2 ), we find that ( x = 0.6 ) and ( y = 0.4 ). Therefore, we can conclude that:

OP⃗=0.6a+0.4b\vec{OP} = 0.6a + 0.4b

Step 3

Process to find \( \vec{ON} \)

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Answer

Let ( \vec{ON} = k \cdot \vec{OP} ). Thus,

ON⃗=k(0.6a+0.4b)\vec{ON} = k(0.6a + 0.4b)

From the ratio we set up using segments, we can say that: ( ON:NB = 3:2 ). Therefore, using the comparison based on the sections derived, we reach:

ONNB=32\frac{ON}{NB} = \frac{3}{2}

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