Inequalities
223. The Perpendicular Promise · Why the shortest path is always straight down
OP is always the shortest distance from O to line AB.
Of all segments joining a point O to a line AB, the perpendicular OP is the shortest. Any slanted segment is the hypotenuse of a right triangle having OP as a leg, and a hypotenuse is always longer than a leg, so the straight-down perpendicular is least.
What this lesson covers
Try to break it
Drag Q along line AB. OP stays perpendicular to AB and stays the shortest segment from O to AB. OQ stretches as Q moves away from P. Try to find a position where OQ is shorter than OP — impossible. The perpendicular wins the shortness contest every time.
How you build it
Construct a perpendicular from external point O.
- Draw a straight line to serve as AB.
- Drop point O above the line, not on it.
- Compass: click O for the centre, then click below line AB so the arc crosses it at two points X and Y.
- Click the left crossing of the arc and line AB to mark X.
- Click the right crossing of the arc and line AB to mark Y.
- Bisector tool: click X, then Y. Because OX = OY, this line passes through O and meets AB at a true right angle.
- Click where the perpendicular meets line AB to mark P.
- Draw the segment from O to P.
The proof, step by step
Prove that the perpendicular is the shortest segment from a point to a line.
- In right angled ΔOPQ, ∠OPQ = 90°.
- ∠OPQ > ∠OQP (Right angle is the greatest angle in a right angled Δ).
- OQ > OP (Side opposite to greater angle is greater).
- ∴ OP is the shortest line drawn from O to AB.
Worked example
A point P is located at a distance of 10 cm from a line l. What is the length of the shortest line segment that can be drawn from P to l?
The shortest distance from a point to a line is the length of the perpendicular segment. Since the distance is given as 10 cm, the perpendicular length is 10 cm.
- 5 cm
- 8 cm
- 10 cm — correct
- 12 cm