Constructions and loci

The perpendicular bisector of a line AB cuts it in half at right angles. Every point on it is the same distance from A as from B.

1. Open your compasses to more than half the length of AB.
2. With the point on A, draw an arc above and below the line.
3. Keep the compasses the same, put the point on B, and draw two arcs that cross the first two.
4. Join the two crossing points with a straight line.

Example: describe how to find the midpoint of a 9 cm line AB without measuring.

1. Construct the perpendicular bisector of AB, as above
2. The midpoint is where it crosses AB, 4.5 cm from each end

An angle bisector cuts an angle exactly in half. Every point on it is the same distance from both arms of the angle.

1. With the compass point on the corner of the angle, draw an arc that crosses both arms.
2. Put the compass point on one crossing, and draw an arc inside the angle.
3. Keeping the compasses the same, do the same from the other crossing, so the arcs meet.
4. Draw a line from the corner through the point where the arcs meet.

Example: an angle of 70° is bisected. What size are the two new angles?

1. The bisector cuts the angle in half: 70 ÷ 2 = 35° each

To draw the shortest line from a point P to a line:

1. With the compass point on P, draw an arc that crosses the line twice.
2. From each crossing point, draw arcs below the line (same radius) so they meet.
3. Draw a line from P through the point where these arcs meet.

This line meets the original line at 90°. It is the shortest distance from P to the line.

Example: why is the perpendicular the shortest route from P to the line?

1. Any other route is the hypotenuse of a right-angled triangle, with the perpendicular as one of its shorter sides
2. The hypotenuse is always the longest side, so the perpendicular is shortest

To construct a triangle from its three sides, you need a ruler and compasses.

1. Draw the longest side with a ruler.
2. Open the compasses to the second length, and draw an arc from one end.
3. Open them to the third length, and draw an arc from the other end.
4. Where the arcs cross is the third corner. Join it to both ends.

Example: describe how to construct a triangle with sides of 7 cm, 6 cm and 5 cm.

1. Draw a 7 cm line AB
2. Draw an arc of radius 6 cm from A, and one of radius 5 cm from B
3. The arcs cross at C; join AC and BC

If the two shorter sides add up to less than the longest, the arcs won't meet: no such triangle exists.

A locus is the set of points that follow a rule. These are the main ones:

Example: a goat is tied to a post with a 3 m rope. Describe the region it can reach.

1. Every point within 3 m of the post
2. A circle of radius 3 m centred on the post, with its inside shaded

"Closer to A than to B" is the side of the perpendicular bisector that contains A.