Geometrical Problems with Coordinates
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=√(x2−x1)2+(y2−y1)2
- Gradient formula: $m=\frac{y_2-y_1}{x_2-x_1}$m=y2−y1x2−x1
- 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
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.
Find the value of $k$k.
Find the area of the triangle.
Question 2
Given Line P: $y=-6x-4$y=−6x−4, Line Q: $y=\frac{x}{6}+6$y=x6+6, Line R: $y=-6x-1$y=−6x−1 and Line S: $y=\frac{x}{6}+1$y=x6+1.
Complete the following:
$m$mP = $\editable{}$
$m$mQ = $\editable{}$
$m$mP x $m$mQ = $\editable{}$
Complete the following:
$m$mQ = $\editable{}$
$m$mR = $\editable{}$
$m$mQ x $m$mR = $\editable{}$
Complete the following:
$m$mR = $\editable{}$
$m$mS = $\editable{}$
$m$mR x $m$mS = $\editable{}$
Complete the following:
$m$mS = $\editable{}$
$m$mP = $\editable{}$
$m$mS x $m$mP = $\editable{}$
What type of quadrilateral is formed by lines: P, Q, R, and S?
Trapezoid
ARectangle
BRhombus
CParallelogram
D