Draw a line l and mark a point P anywhere outside the line. Construct a perpendicular to the given line l through P.
To construct a perpendicular to line l through point P (outside the line): (1) With P as centre and any radius, draw an arc that intersects line l at two points A and B. (2) With A as centre and radius greater than half of AB, draw an arc. (3) With B as centre and the same radius, draw another arc intersecting the previous arc at Q. (4) Join PQ. PQ is perpendicular to line l. This works because P and Q are both equidistant from A and B, so they lie on the perpendicular bisector of AB, which is perpendicular to l.
Explanation
The chapter provides a hint: find a line segment on l whose perpendicular bisector passes through P. When we draw an arc from P intersecting l at A and B, P is equidistant from A and B (by construction). Finding another point Q equidistant from A and B gives us two points on the perpendicular bisector of AB. Since AB lies on line l, its perpendicular bisector is perpendicular to l and passes through P.
Solution Steps
Step 1: Draw line l and mark point P outside it.
Step 2: With P as centre, draw an arc intersecting l at points A and B.
Step 3: With A as centre and radius > AB/2, draw an arc above or below l.
Step 4: With B as centre and same radius, draw an arc intersecting the previous arc at Q.
Step 5: Join PQ. This is the required perpendicular to l through P.