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Problem-solving

9.5 Coordinate Geometry

Show all working. Partial marks are given for method.

  1. 1
    **Line through two points.** The line AB passes through $A(-2, 1)$ and $B(7, 4)$. (a) Find the gradient as a fraction in simplest form. (b) Find the equation in $y = mx + c$ form. (c) State the equation of a line parallel to AB through the origin.

    Working space

  2. 2
    **Dividing a segment.** Point C divides AB in ratio 2:1, where $A(-2, 1)$, $B(7, 4)$. Find C.

    Working space

  3. 3
    **Distance and midpoint together.** Two villages at $A(2, 5)$ and $B(10, 11)$ on a map (km). (a) Find the straight-line distance. (b) Find the midpoint. (c) A new road perpendicular to AB through the midpoint. Find its equation.

    Working space

  4. 4
    **Mid-segment investigation.** A quadrilateral with vertices A(0, 0), B(6, 0), C(8, 4), D(2, 6). (a) Find the midpoints P, Q, R, S of sides AB, BC, CD, DA. (b) Show PQRS is a parallelogram by comparing gradients.

    Working space

  5. 5
    **Perpendicular bisector.** Find the equation of the perpendicular bisector of A(1, 2) and B(5, 6).

    Working space

  6. 6
    **Triangle on coordinate plane.** Find the perimeter of the triangle with vertices A(0, 0), B(4, 3), C(8, 0).

    Working space

  7. 7
    **Distance and perpendicular distance.** A point $P(5, 1)$ and a line $y = 2x - 1$. (a) Find the equation of the line through P perpendicular to the given line. (b) Find the foot of the perpendicular (i.e. the point on the line closest to P). (c) Find the perpendicular distance from P to the line.

    Working space

  8. 8
    **Investigating parallel/perpendicular.** Lines $\ell_1: y = mx + 2$ and $\ell_2: y = (3 - m)x + 5$. (a) For what value of $m$ are $\ell_1$ and $\ell_2$ parallel? (b) For what value(s) of $m$ are they perpendicular?

    Working space

  9. 9
    **Triangle bounded by three lines.** Sketch the triangle bounded by $y = x + 1$, $y = -x + 5$, $y = 0$. (a) Find the three vertices. (b) Find the area.

    Working space

  10. 10
    **Identifying a parallelogram.** Show that the points A(1, 1), B(4, 1), C(6, 4), D(3, 4) form a parallelogram.

    Working space

  11. 11
    **Centroid of a triangle.** A triangle has vertices A(0, 0), B(6, 0), C(3, 6). (a) Find the centroid (average of vertices). (b) Find the medians and verify they all pass through the centroid.

    Working space

  12. 12
    **Reflection of a line.** The line $y = 2x + 1$ is reflected in the $x$-axis. (a) Find the equation of the image. (b) Find the equation when reflected in the $y$-axis.

    Working space