topic badge
India
Class XI

Geometrical Problems with Coordinates

Lesson

We've looked at how to plot straight lines on the number plane. Now we are going to look at how to plot a series of coordinates to create geometric shapes.

Here are some helpful formulae and properties that will help us solve these kinds of problems:

  • Distance formula: $d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$d=(x2x1)2+(y2y1)2
  • Gradient formula: $m=\frac{y_2-y_1}{x_2-x_1}$m=y2y1x2x1
  • Mid-point formula: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$(x1+x22,y1+y22)
  • Parallel lines have equal gradient: $m_1=m_2$m1=m2
  • The product of the gradients of perpendicular lines is $-1$1: $m_1m_2=-1$m1m2=1
Remember!

Different shapes have different properties.

These can be used to help plot and identify features of shapes on a number plane, so make sure you're familiar with the properties of different triangles and quadrilaterals.

 

Worked Examples

Question 1

$A$A$\left(-2,-1\right)$(2,1), $B$B$\left(0,0\right)$(0,0) and $C$C$\left(1,k\right)$(1,k) are the vertices of a right-angled triangle with right angle at $B$B.

  1. Find the value of $k$k.

  2. Find the area of the triangle.

Question 2

Given Line P: $y=-6x-4$y=6x4, Line Q: $y=\frac{x}{6}+6$y=x6+6, Line R: $y=-6x-1$y=6x1 and Line S: $y=\frac{x}{6}+1$y=x6+1.

  1. Complete the following:

    $m$mP = $\editable{}$

    $m$mQ = $\editable{}$

    $m$mP x $m$mQ = $\editable{}$

  2. Complete the following:

    $m$mQ = $\editable{}$

    $m$mR = $\editable{}$

    $m$mQ x $m$mR = $\editable{}$

  3. Complete the following:

    $m$mR = $\editable{}$

    $m$mS = $\editable{}$

    $m$mR x $m$mS = $\editable{}$

  4. Complete the following:

    $m$mS = $\editable{}$

    $m$mP = $\editable{}$

    $m$mS x $m$mP = $\editable{}$

  5. What type of quadrilateral is formed by lines: P, Q, R, and S?

    Trapezoid

    A

    Rectangle

    B

    Rhombus

    C

    Parallelogram

    D

Outcomes

11.CG.SL.1

Brief recall of 2D from earlier classes. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axes, point-slope form, slope-intercept form, two-point form, intercepts form and normal form. General equation of a line. Distance of a point from a line.

What is Mathspace

About Mathspace