Points, lines and circles: relative positions

What is it about?

In the coordinate plane a point is given by two numbers. The horizontal axis is the x-axis, the vertical one the y-axis; they meet at the origin $O(0, 0)$ . For $P(3, 5)$ the x-coordinate comes first: from the origin go 3 to the right, then 5 up. Swapping the numbers gives a different point, $(5, 3)$ .

Name	Endpoints	Length
segment AB	A and B	can be measured
ray AB	only A (starting point)	endless
line AB	none	endless

Two lines $g$ and $h$ that meet at a right angle are perpendicular ( $g \perp h$ ); lines that never meet are parallel ( $g \parallel h$ ). With a set square: put the edge on $g$ . If $h$ lies on the centre line, $h \perp g$ ; if $h$ runs along one of the thin lines parallel to the edge, $h \parallel g$ .

The distance of a point $P$ from a line $g$ is the length of the perpendicular from $P$ to $g$ – the shortest path.

A circle consists of all points at the same distance $r$ (the radius) from its centre $M$ ; the diameter is $d = 2 \cdot r$ . Comparing the distance from $M$ to a line with $r$ tells you whether the line misses the circle (greater), touches it at one point – a tangent – (equal), or cuts it at two points – a secant – (smaller). A tangent is perpendicular to the radius at the point of contact.

Try it

Drag A and B and read off their coordinates using the dashed guide lines.
Switch between segment, ray and line. Load a line
Measure a slanted segment with the ruler.
Check two perpendicular lines with the set square, then drag D away.
Show the distance of a point and drag P.
Increase the radius of a circle from 2 cm to 4 cm. When is g a tangent?
Play the point game.

Tasks

Task 1: Completing a square

Plot $A(1, 2)$ , $B(6, 2)$ and $C(6, 7)$ . Which point $D$ completes the square $ABCD$ ?

Show solution

$D(1, 7)$ . Opposite sides are parallel, neighbouring sides are perpendicular. Show

Task 2: Finding a perpendicular

Load task

The line $g$ passes through $A(2, 2)$ and $B(8, 5)$ . Find a grid point $D$ such that the line through $C(4, 7)$ and $D$ is perpendicular to $g$ .

Show solution

Along $g$ : 2 right, 1 up. A perpendicular goes 1 right, 2 down: $D(5, 5)$ , $D(6, 3)$ or $D(7, 1)$ . Show

Task 3: Parallel and distance

The line $g$ passes through $A(1, 2)$ and $B(9, 2)$ . (a) Draw the parallel $h$ to $g$ through $C(2, 6)$ . (b) How far apart are $g$ and $h$ ? (c) How far is the point $P(5, 7)$ from $g$ ?

Show solution

(a) $h$ is horizontal through $C$ , e.g. through $D(8, 6)$ . (b) $4\,\text{cm}$ (from $y = 2$ to $y = 6$ ). (c) The perpendicular from $P$ meets $g$ at $F(5, 2)$ , so the distance is $5\,\text{cm}$ . Check

Task 4: Tangents

A circle has centre $M(5, 4)$ and radius $3\,\text{cm}$ . Give two tangents that are parallel to the x-axis.

Show solution

The horizontal lines through $(0, 1)$ and $(0, 7)$ , touching the circle at $(5, 1)$ and $(5, 7)$ . Show

Ideas for the classroom

Use: grade 5, introducing the coordinate plane (first quadrant), parallel, perpendicular, distance and tangent; the point game works well as a warm-up.
Model: all points lie on grid points; one unit is 1 cm. Perpendicular, parallel and tangent are decided exactly; lengths are rounded to millimetres like a ruler reading. The set square is shown at half size.
Common stumbling blocks: swapping x and y (the game detects this), “perpendicular” meaning “vertical”, measuring a distance along a slanted line instead of at right angles.