Constructions and Tilings · Lesson 2 of 12
Construction of a 90° Angle at a Given Point
“Construct a perpendicular exactly where it is needed and discover how a taut rope can carry the same geometry.”
• Make a specified point the midpoint of a chosen segment. • Construct a 90° angle on a line using a compass. • Explain why one new arc intersection is enough. • Relate rope constructions to equal-distance reasoning.
Making the Given Point a Midpoint
You may need a right angle at a particular point O on a line: for a design, a boundary, or another construction. Drawing a perpendicular bisector of an arbitrary segment on that line could put the right angle in the wrong place. We therefore choose the segment so that its midpoint is already O.
Put the compass point at O and choose a convenient radius. With this unchanged opening, cut the line on opposite sides of O at X and Y. Both OX and OY equal the chosen radius, so O is the midpoint of XY. If the original line drawing is too short, extend it on either side first.
Now the perpendicular bisector of XY must pass through O. We have converted the new task into a construction we already understand. The length XY is not prescribed; we are free to choose a length that makes the drawing convenient.
Constructing the Perpendicular
Because the midpoint is known, the usual method can be shortened. We need another point on the perpendicular bisector, and then the straight line joining it to O. This explains the shortcut rather than asking you to remember a different rule.
- Mark X and Y on opposite sides of O with OX = OY.
- Open the compass to more than half XY. Draw equal-radius arcs from X and Y above the line, meeting at A.
- Draw the ray OA. Both O and A lie on the perpendicular bisector of XY.
- The angle between OA and either direction of the original line is 90°.
Why does A lie on the right line? The two arcs give AX = AY. A point equally far from X and Y belongs to their perpendicular bisector. O also belongs to it because OX = OY. A line is determined by two distinct points, so OA is that bisector.
The method does not depend on the direction of the line. Turn the page and the distance relationships remain the same. A perpendicular to a sloping line is usually sloping too; vertical appearance is not the definition of perpendicular.
Problem
Construct a 90° angle at O on a horizontal line.
- 1.Mark X and Y 2 cm from O in opposite directions, so XY = 4 cm.
- 2.Choose radius 3 cm for the arcs centred at X and Y. This is greater than half XY, which is 2 cm.
- 3.Label their intersection above the line A and join OA.
- 4.The line through O and A is the perpendicular bisector of XY. Therefore ∠AOX and ∠AOY are both 90°.
Problem
Use the same idea when the line rises from left to right.
- 1.Mark O on the sloping line and use one compass opening to place X and Y on opposite sides.
- 2.Draw equal arcs from X and Y on a convenient side. The arcs need not be above the page; they need only meet on one side of the given line.
- 3.Join the intersection to O. Equal distances identify the perpendicular bisector regardless of orientation.
- 4.If a vertical line does not make 90° with the given line, it is not the required perpendicular.
Do not confuse a perpendicular through O with a perpendicular bisector of the entire visible drawing. The visible ends may be at unequal distances from O. We deliberately create the auxiliary segment XY with midpoint O.
Construction Methods in Śulba-Sūtras
Geometric constructions were used long before the modern school compass. The Śulba-Sūtras are ancient Indian texts concerned with geometric methods for constructing ritual fire altars. They include ways to construct perpendiculars and perpendicular bisectors using ropes.
A taut rope can provide a straight direction. If one end stays fixed and the other end moves while the rope length stays constant, the moving end traces a circle or an arc. The fixed length plays the role of a compass radius. This allows a geometric idea learned on paper to be used on a larger surface.
For a perpendicular bisector, place pegs at X and Y. Take a rope whose usable length is greater than XY. Fold it in half and mark its midpoint, ignoring the portions used for fastening. Attach its ends to the pegs. Pull the midpoint to one side until both halves are taut, and mark that point A. Repeat on the other side to obtain B. A taut half has the same length on both sides, so AX = AY and BX = BY. Joining A and B gives the perpendicular bisector.
A Rope as a Geometric Tool
The important part of the rope method is not its material but the equality it preserves. A stretched rope whose two halves have equal length creates an equidistant point. You can now reuse that observation to construct a right angle at a known midpoint.
Problem
X and Y are equally far from O on a straight line. A rope midpoint is pulled taut to A. Why is OA perpendicular?
- 1.The chosen segment has OX = OY, so O lies on its perpendicular bisector.
- 2.The rope halves give AX = AY, so A lies on the same bisector.
- 3.O and A are distinct points, and their line is therefore the perpendicular bisector.
- 4.OA meets XY at 90°. No length measurement of the angle is needed.
Use string and two fixed points on a board or sheet. Keep the usable halves equal and taut, then compare the resulting perpendicular direction with a paper compass construction. Small fastening or stretching errors affect the practical drawing, even though the geometric reasoning is exact.
Check Your Understanding
Use these questions to check the conditions behind each method, as well as the result. For a diagram-based question, follow the labels and given information rather than judging by appearance.
Quiz
Why do we first choose OX = OY on opposite sides of O?
Why is one new intersection A sufficient?
What condition puts A on the perpendicular bisector of XY?
What acts as a compass radius in a rope construction?
What remains true if the given line is rotated?
Practice Problems
- Construct a perpendicular at O on a line of your choice. Show the equal lengths used.
- Explain why a lower pair of arcs is unnecessary when O is already the midpoint.
- For XY = 6 cm with midpoint O, choose a radius for the arcs from X and Y and justify it.
- A rope has unequal usable halves when pulled to A. Is AX = AY guaranteed? Explain.
- Describe how to construct a perpendicular at a known midpoint using a rope and pegs.
- Explain why joining any arbitrary point above a line to O does not necessarily make a right angle.
Mark X and Y at equal distances from O on opposite sides. Draw equal-radius arcs with radius greater than OX, and join their intersection A to O.
Key Takeaways
• Make the specified point a midpoint before constructing the perpendicular. • The auxiliary segment can be chosen to suit the drawing. • One arc intersection and the known midpoint determine the bisector line. • Perpendicular means 90°, regardless of page orientation. • Equal taut rope lengths reproduce equal compass radii. • The rope method works because it locates equidistant points.