Construct a right angle to a line

1 · Learn

A perpendicular meets a line at 90°. There are two different starting tasks: the given point may lie ON the line or OFF it. Use a long enough straight line and extend it when necessary.

At a point P ON the line: use one compass opening centred at P to mark A and B on opposite sides of P. PA = PB, so P is their midpoint. From A and B use equal radii greater than half of AB to make crossing arcs at Q above the line. Join P to Q. This is the perpendicular bisector of AB, so it is perpendicular at P.

From a point P OFF the line: choose a compass radius greater than P's shortest distance to the line. An arc centred at P then cuts the line twice, at A and B. PA = PB. From A and B draw equal-radius arcs meeting at Q on the other side of the line from P. Join P and Q. Both points are equally distant from A and B, so PQ is their perpendicular bisector. Label its foot H on the original line.

PH is the shortest distance from P to the line. For any other point R on the line, triangle PHR is right-angled: PR² = PH² + HR². Since HR² is positive when R differs from H, PR is longer than PH. A sloping segment does not give the point-to-line distance.

2 · Worked example

In the diagram P is above a horizontal line, with H directly below it. PH = 3 cm and HR = 4 cm. The sloping distance PR is 5 cm because 3² + 4² = 5². The distance from P to the line is 3 cm, not 5 cm.

3 · Your turn

Construct a perpendicular at a point on a line, then from a point 4 cm above another horizontal line. For the second diagram mark R 3 cm from the foot H. Find PH and PR and identify the point-to-line distance.

4 · Apply your learning

Leave your construction arcs visible and label the equal radii used at each stage. Compare the two starting situations: a diagram for a point on the line is not a construction from a point outside it.