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Geometric Problem Solving - Higher


Geometric problems can be solved using the rules for adding and subtracting vectors and multiplying vectors by a scalar.

Example:
OABC is a parallelogram. Geometric Problem Solving | Mathematics for Grade 10 is represented by the vector a and Geometric Problem Solving | Mathematics for Grade 10 is represented by the vector c.
M is the mid-point of BC and N is the point on OB such that ON:NB = 2:1.

  • Find expressions for these vectors, giving the answers in the simplest form: Geometric Problem Solving | Mathematics for Grade 10
  • Show that points A, N and M lie on a straight line.
    Geometric Problem Solving | Mathematics for Grade 10

Part 1:
Point N is 2/3 of the distance from O to B.
Geometric Problem Solving | Mathematics for Grade 10
Point M is half the distance from C to B.
Geometric Problem Solving | Mathematics for Grade 10
Geometric Problem Solving | Mathematics for Grade 10

Part 2: 
To prove that points A, N and M lie on a straight line:
Show Geometric Problem Solving | Mathematics for Grade 10 is a multiple of Geometric Problem Solving | Mathematics for Grade 10
Geometric Problem Solving | Mathematics for Grade 10
Therefore Geometric Problem Solving | Mathematics for Grade 10 are parallel. They also share a common point A so they lie on the same straight line.
Points which lie on the same straight line are called collinear.

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