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OYZ is a parallelogram - Edexcel - GCSE Maths - Question 1 - 2019 - Paper 1
Question 1
OYZ is a parallelogram. \( ightarrow OX = a\) \( ightarrow OY = b\) P is the point on \(OX\) such that \(OP : PX = 1 : 2\) R is the point on \(OY\) such that \(OR ... show full transcript
Worked Solution & Example Answer:OYZ is a parallelogram - Edexcel - GCSE Maths - Question 1 - 2019 - Paper 1
Step 1
Find point P on OX such that OP : PX = 1 : 2
Answer
Let (OP = x) and (PX = 2x). Thus, the total length (OX = OP + PX = x + 2x = 3x). Since (OX = a), we can express (x) as:
Therefore, the position vector of point (P) is given by:
(\overrightarrow{OP} = \frac{1}{3}\overrightarrow{OX} = \frac{1}{3}a)
Step 2
Step 3
Calculate ZP and ZR in terms of a and b
Answer
Point (Z) can be expressed in terms of both points (P) and (R):
(\overrightarrow{ZP} = \overrightarrow{OZ} - \overrightarrow{OP} = \overrightarrow{OY} - \overrightarrow{OP} = b - \frac{1}{3}a) (\overrightarrow{ZR} = \overrightarrow{OZ} - \overrightarrow{OR} = \overrightarrow{OX} - \overrightarrow{OR} = a - \frac{1}{4}b)
Step 4
Express ZP and ZR as multiples of the same vector
Answer
Using the previously derived vectors:
(\overrightarrow{ZP} = b - \frac{1}{3}a) and (\overrightarrow{ZR} = a - \frac{1}{4}b).
To express them in a ratio, we set:
(ZP : ZR = (b - \frac{1}{3}a) : (a - \frac{1}{4}b)).
This ratio can be simplified further using common denominators and combining the terms.