GCSE OCR Maths Vectors: ABC is a parallelogram. BD → =

ABC is a parallelogram - OCR - GCSE Maths - Question 16 - 2017 - Paper 1

Question 16

ABC is a parallelogram. BD → = a and AD → = b. F is the midpoint of BC. G is the midpoint of DC. AE → = 3EB →. (a) Write down simplified expressions in terms o... show full transcript

Worked Solution & Example Answer:ABC is a parallelogram - OCR - GCSE Maths - Question 16 - 2017 - Paper 1

Step 1

Write down simplified expressions in terms of a and b for (i) AB →

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Answer

To find the vector AB →, we can express it in terms of the vectors a and b. Since AB → is the diagonal from point A to point B in the parallelogram, we have:

$AB → = AD → + DB →$

Substituting the given vectors, we get:

$AB → = b + a$

Step 2

Write down simplified expressions in terms of a and b for (ii) EB →

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Answer

To find the vector EB →, we start from the midpoint F and notice that E is half of the way from B towards A:

Since F is the midpoint of BC, we can write:

$EB → = EF → + FB →$

Substituting the expressions we know leads us to:

$EB → = \frac{1}{2}AB → = \frac{1}{2}(b + a)$

Step 3

Show that EF → = \frac{1}{4}(3b - a)

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Answer

To prove that EF → equals \frac{1}{4}(3b - a), we can express EF in terms of the vector definitions. Given the positions of E and F as midpoints, we have:

Using the vectors:

$EF → = EB → + BF →$

Substituting:

$EF → = \frac{1}{2}(b + a) + \frac{1}{2}BB →$

After simplification, we find:

$EF → = \frac{1}{4}(3b - a)$

Step 4

Prove that EF → and AG → are parallel.

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Answer

To show that EF → and AG → are parallel, we analyze their directional vectors. We know:

From AG:

$AG → = AE → - EF →$

and since we have already expressed EF → in terms of a and b:

We can arrange it as:

$AG → = 2EF →$

This indicates that AG → is a scalar multiple of EF →, confirming that they are parallel.

ABC is a parallelogram - OCR - GCSE Maths - Question 16 - 2017 - Paper 1

Worked Solution & Example Answer:ABC is a parallelogram - OCR - GCSE Maths - Question 16 - 2017 - Paper 1

Write down simplified expressions in terms of a and b for (i) AB →

Write down simplified expressions in terms of a and b for (ii) EB →

Show that EF → = \frac{1}{4}(3b - a)

Prove that EF → and AG → are parallel.

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