Construct tangents to the circle from point B with radius 3.5 cm and centre A. Point B is at a distance 8.5 cm from the centre.

 Steps of construction:

1. Draw a circle with centre ‘A’ and radius 3.5 cm . Take a point B such that AB = 8.5 cm.
2. Obtain the midpoint M of seg AB. (8.5/2=4.25 cm) by using scale or by using perpendicular bisector method.
3. Draw a circle with M as centre and BM as radius.
4. Let C and D be the points of intersection of these two circles.
5. Draw rays BC and BD which are the required tangents.

To draw Tangent to a given circle at any point on it

 1) With centre ‘O’, draw a circle of any radius, and mark a point ‘P’ on it.

2) Draw a line joining O and P.

3)  Extend this line OP to Q so that OP=PQ.

4) Take any convenient radius and O as centre draw an arc as shown below,

5)  By taking same radius and Q as centre draw another arc cutting previously drawn arc as shown

6) Draw a line through P and R. Then this line is the required Tangent.

Question

1) Construct a tangent at any point P on a circle of radius 3 cm.

2) Draw a circle of radius 4 cm and construct a tangent at any point on the circle.

 

To draw a continues curve of circular arcs passing through any number of parts not in straight lines

 1) Let us consider 3 points A, B, C be given points which are not present in straight line. 

2) Draw lines connecting A with B, B with C.

3) Draw perpendicular bisectors to lines AB, BC

4) These two perpendicular bisectors intersect at point ‘O’

5) With O as centre and radious equal to OA, draw an arc passing through points A, B, C

TO FIND CENTRE OF AN ARC

Method 2:

Problem: To find the centre of an Arc

1. Draw an arc AB.

2. Draw a line (Chord) that connects the two points of the arc.

3. Set your compass more than half of the length of line and place compass on one end of the line ( or arc) and draw arc's above and below the arc AB.

4. Keeping the same distance set on your compass, swing arcs from another end of the line, draw arcs above and below the arc AB cutting previously drawn arc's.

5. Draw a line through the two intersection points of the arcs.

5. It will cut the line AB at 'O', This is the centre of the chord.

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).