6.3 Geometry  (Page 5/7)

and

It can be seen that $PS=QS$ as expected.

The following video provides a summary of the midpoint of a line.

Co-ordinate geometry

1. In the diagram given the vertices of a quadrilateral are F(2;0), G(1;5), H(3;7) and I(7;2).

   

   1. What are the lengths of the opposite sides of FGHI?
   2. Are the opposite sides of FGHI parallel?
   3. Do the diagonals of FGHI bisect each other?
   4. Can you state what type of quadrilateral FGHI is? Give reasons for your answer.
2. A quadrialteral ABCD with vertices A(3;2), B(1;7), C(4;5) and D(1;3) is given.
   1. Draw the quadrilateral.
   2. Find the lengths of the sides of the quadrilateral.
3. ABCD is a quadrilateral with verticies A(0;3), B(4;3), C(5;-1) and D(-1;-1).
   1. Show that:
      1. AD = BC
      2. AB $\parallel$ DC
   2. What name would you give to ABCD?
   3. Show that the diagonals AC and BD do not bisect each other.
4. P, Q, R and S are the points (-2;0), (2;3), (5;3), (-3;-3) respectively.
   1. Show that:
      1. SR = 2PQ
      2. SR $\parallel$ PQ
   2. Calculate:
      1. PS
      2. QR
   3. What kind of a quadrilateral is PQRS? Give reasons for your answers.
5. EFGH is a parallelogram with verticies E(-1;2), F(-2;-1) and G(2;0). Find the co-ordinates of H by using the fact that the diagonals of a parallelogram bisect each other.
   

Transformations

In this section you will learn about how the co-ordinates of a point change when the point is moved horizontally and vertically on the Cartesian plane. You will also learn about what happens to the co-ordinates of a point when it is reflected on the $x$ -axis, $y$ -axis and the line $y=x$ .

Translation of a point

When something is moved in a straight line, we say that it is translated . What happens to the co-ordinates of a point that is translated horizontally or vertically?

Discussion : translation of a point vertically

Complete the table, by filling in the co-ordinates of the points shown in the figure.

What do you notice about the $x$ co-ordinates? What do you notice about the $y$ co-ordinates? What would happen to the co-ordinates of point A, if it was moved to the position of point G?

When a point is moved vertically up or down on the Cartesian plane, the $x$ co-ordinate of the point remains the same, but the $y$ co-ordinate changes by the amount that the point was moved up or down.

For example, in [link] Point A is moved 4 units upwards to the position marked by G. The new $x$ co-ordinate of point A is the same ( $x$ =1), but the new $y$ co-ordinate is shifted in the positive $y$ direction 4 units and becomes $y$ =-2 +4 =2. The new co-ordinates of point A are therefore G(1;2). Similarly, for point B that is moved downwards by 5 units, the $x$ co-ordinate is the same ( $x=-2,5$ ), but the $y$ co-ordinate is shifted in the negative $y$ -direction by 5 units. The new $y$ co-ordinate is therefore $y$ =2,5 -5 =-2,5.