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.

Constructing the perpendicular from a point to a line

           When two lines intersect to form right angles then such lines are known as Perpendicular to each other. Perpendicular lines are Co-Planar (i.e., They lie in same plane) and intersect at right angle (90°).

Method 1: Constructing a perpendicular to a line through a point on it

1) Assume given line as AB and a point P is present near middle of the line on it,

2) Taking P as a center and any suitable radius on compass, draw an arc cutting line segment AB at two distinct points (C, D) as shown below

3) Taking C as centre and a suitable radius (more than CP or DP distance) draw an arc above the line segment, with D as centre and the same radius as previous, draw an arc cutting previously drawn arc,

4) Mark the intersecting point as E.

5) Join the points E and P as shown, and the line segment EP is the required perpendicular to line AB through point P.

Note: This method we follow when we don't have Protractor with us.

Manually Bisecting a Circular Arc

          In Geometry, bisecting an arc is cutting arc exactly in half.

Problem: To bisect 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.

Bisecting an Angle

           An angle bisector divides an angle into equal angles. If the angle is 'θ', the two angles made after bisecting will be 'θ/2'. Angle bisectors can be constructed for an acute angle, obtuse angle, or a right angle too. This angle bisector passes through the vortex of an angle, as shown in the figure.

Problem: To bisect 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. Place your compass on one of the intersection points (i.e., D) and draw an arc inside the angle. Do not adjust the compass,

4. Place your compass on the other intersection point (i.e., E) and draw an arc that cuts the previously drawn arc as shown,

5. Mark the intersecting point of arc's as F. Finally, draw a line connecting the vortex B and point F. This is the angular bisector of our angle ∠ABC.

6. You can measure both sides of the angle to check they are equal.

 

Bisecting a line

          In Geometry, Bisecting a line is cutting a line exactly in half. It may also be referred to as constructing a perpendicular bisector as the line you are drawing will be at a right angle to the original line. A line that passes through the mid point of the line segment is known as the segment Bisector.

Problem: To bisect a given straight line AB

1. Draw a straight line of given dimension and mark start and end points as A, B.

2.  To bisect this line, set your compass to just more than half the length of the line. Keep your compass this size for the rest of the question.

3. Place compass on one end of the line (i.e., point B) and draw an arc that crosses the line AB. Do not adjust the compass and keep the same length.

4. Place compass on the other end of the line (i.e., point A) and draw an arc. It must cut the other arc above and below the line. Mark the point of intersections as C,D.

5. Draw a straight line through where the two arcs intersect above and below the line (i.e., C,D). It will cut the line AB at O. CD bisects AB at right angles. 

6. Measure both sides of the line from 'O' check that they are of equal length, if they are equal we did it correctly.