60. The Perpendicular Drop · constructing right angles from outside

Try to break it

What happens if C moves closer to AB? Try dragging it down!

How you build it

Construct a perpendicular from an external point.

• Place point A — one end of the line.
• Place point B — the other end of the line.
• Draw the straight line through A and B.
• Place point C above line AB — the external point.
• With the compass on centre C, open it wide and draw an arc that cuts line AB at two points.
• Mark point P where the arc crosses line AB on the left.
• Mark point Q where the arc crosses line AB on the right.
• Keeping the same radius, draw an arc with centre P below line AB.
• With the same radius, draw an arc with centre Q so it crosses the arc from P at point D.
• Mark point D where the two arcs cross, below line AB.
• Draw the line through C and D — it passes through M and is the perpendicular from C to line AB.

The proof, step by step

Prove that CM is perpendicular to AB.

• CP = CQ because they are radii of the same arc drawn from C.
• DP = DQ because they are radii of equal arcs drawn from P and Q.
• Therefore, both C and D are equidistant from P and Q.
• The line joining two points equidistant from the endpoints of a segment is its perpendicular bisector.
• Hence, CD is perpendicular to AB at M.

Worked example

In the construction of a perpendicular from an external point C to line AB, arcs are drawn from P and Q with equal radii. What is the primary reason for using equal radii?

Equal radii from P and Q ensure DP = DQ. Combined with CP = CQ, this proves that both C and D lie on the perpendicular bisector of PQ, making CD ⊥ AB.

• To ensure the arcs intersect at a single point D
• To make DP equal to DQ, so D lies on the perpendicular bisector of PQ — correct
• To make the construction faster and easier to draw
• To ensure that angle CPQ equals angle CQP