8.3 Summary & Exercises

Analytical geometry; summary and exercises

• Figures can be represented on the Cartesian plane
• The formula for finding the distance between two points is:
  $\mathrm{Distance}=\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2}$
• The formula for finding the gradient of a line is:
  $\mathrm{Gradient}=\frac{y_2-y_1}{x_2-x_1}$
• The formula for finding the midpoint between two points is:
  $S\left(\frac{x_1+x_2}{2},;,\frac{y_1+y_2}{2}\right)$
• If two lines are parallel then they will have the same gradient, i.e. $m_{\mathrm{AB}}=m_{\mathrm{CD}}$ . If two lines are perpendicular than we have: $-\frac{1}{m_{\mathrm{AB}}}=m_{\mathrm{CD}}$

End of chapter exercises

1. Represent the following figures on the Cartesian plane:
   1. Triangle DEF with D(1;2), E(3;2) and F(2;4)
   2. Quadrilateral GHIJ with G(2;-1), H(0;2), I(-2;-2) and J(1;-3)
   3. Quadrilateral MNOP with M(1;1), N(-1;3), O(-2;3) and P(-4;1)
   4. Quadrilateral WXYZ with W(1;-2), X(-1;-3), Y(2;-4) and Z(3;-2)
2. 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.
3. 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.
4. 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 $âˆ\yen$ DC
   2. What name would you give to ABCD?
   3. Show that the diagonals AC and BD do not bisect each other.
5. 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 $âˆ\yen$ PQ
   2. Calculate:
      1. PS
      2. QR
   3. What kind of a quadrilateral is PQRS? Give reasons for your answers.
6. 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.
   
7. PQRS is a quadrilateral with points P(0; −3) ; Q(−2;5) ; R(3;2) and S(3;–2) in the Cartesian plane.
   1. Find the length of QR.
   2. Find the gradient of PS.
   3. Find the midpoint of PR.
   4. Is PQRS a parallelogram? Give reasons for your answer.
8. A(–2;3) and B(2;6) are points in the Cartesian plane. C(a;b) is the midpoint of AB. Find the values of a and b.
   
9. Consider: Triangle ABC with vertices A (1; 3) B (4; 1) and C (6; 4):
   1. Sketch triangle ABC on the Cartesian plane.
   2. Show that ABC is an isoceles triangle.
   3. Determine the co-ordinates of M, the midpoint of AC.
   4. Determine the gradient of AB.
   5. Show that the following points are collinear: A, B and D(7;-1)
10. In the diagram, A is the point (-6;1) and B is the point (0;3)

    

    1. Find the equation of line AB
    2. Calculate the length of AB
    3. A’ is the image of A and B’ is the image of B. Both these images are obtain by applying the transformation: (x;y) $→$ (x-4;y-1). Give the coordinates of both A’ and B’
    4. Find the equation of A’B’
    5. Calculate the length of A’B’
    6. Can you state with certainty that AA'B'B is a parallelogram? Justify your answer.