Midpoints, parallel lines and vector arguments.
Example 1
In triangle OAB, M and N are the midpoints of OA and OB. If OA=a and OB=b, which identity proves MN is parallel to AB?
Change the shape while M and N remain midpoints. The diagram illustrates the claim; an algebraic argument proves it.Show worked answer
Answer: MN=2b−a
- OM=2a and ON=2b.
- MN=ON−OM=2b−a=2AB, a scalar multiple of AB.
Example 2
A=(0,0),B=(8,5) and C=(24,15). Which relation proves they are collinear?
Compare the position vectors from A. Their directions can support your algebraic argument.Show worked answer
Answer: AC=3AB
- AC=(24,15)=3AB.
- Vectors from A are parallel, so A,B and C lie on one line.
Example 3
A=(0,0),B=(10,7) and C=(20,14). Which relation proves they are collinear?
Compare the position vectors from A. Their directions can support your algebraic argument.Show worked answer
Answer: AC=2AB
- AC=(20,14)=2AB.
- Vectors from A are parallel, so A,B and C lie on one line.