TO FIND CENTRE OF AN ARC

Method 1:

1. Draw an Arc AB

2. Draw any two chords on arc as CD, EF

3. Draw Bisector to CD, 

4. Draw Bisector to EF, 

5. Extend these bisectors until they meet at a point.

6. Mark this point as 'O'.

7. This point 'O' is the centre of the arc.

TRISECTING A Right angle

           An angle Trisector divides an angle into three equal angles. If the angle is 'θ', the three angles made after trisecting will be 'θ/3'.

Problem: To Trisect a Right Angle ABC

1. Let us consider ∠ABC is right angle, B as vortex,

2. Take a compass and draw an arc cutting Horizontal and vertical lines at D and E as shown.

3. With same radious on compass from point E scribe an arc that cuts previously drawn arc at F, and from point D scribe an arc and it will cut the previously drawn arc at point G. 

4. Then join the points BF, BG 

5. These will form the trisector of the given right angle.

Trisecting an angle

          An angle Trisector divides an angle into three equal angles. If the angle is 'θ', the three angles made after trisecting will be 'θ/3'.

Problem: To Trisect a given Angle ABC

1. Let us consider some angle ∠ABC, B as vortex,

2. Take some length in compass and place your compass on the vertex of the angle, here on point B. Draw an arc that crosses both sides of the angle, lines AB and BC. Mark those points as D,E as shown,

3. Connect D and E with scale, 

4. Mark centre point of DE line (or Draw an angular bisector) mark it as 'T', with TD or TE length draw an arc as 'T' as centre.

5. With the same length draw two arcs from point D and E on previously drawn arc.

6. Mark the point of intersections as F, G.

7. Connect FB and GB points with scale, these will form an Angular Trisector of previously given angle.

TO DIVIDE A LINE

To divide a line segment into equal number of parts using a compass only, Example Divide the line into 5 parts,

Steps of construction:

1) Draw a line segment AB of any length.

2) Draw any ray AX, making an acute angle (angle less than 90°) with AB.

3) Now take a compass and with any length draw an arc cutting the line ray AX at point 1, mark this point as A1.

4) With the length A1, Draw 4 more arcs with centres 1, 2, 3, 4 as shown and mark those intersecting points as A2, A3, A4, A5 as shown below.

AA1 = A1A2 = A2A3 = A3A4 = A4A5

5) Now join BA5, it will form a triangle, similarly draw lines from the remaining 4 arcs parallel to line BA5 as shown.

6) The above points divide the line AB into equal parts (You can cross check by measuring the distances).