A. External Tangents
Draw the given circles with centres A and B
- Draw a line segment joining A and B
- At A and B construct perpendiculars to AB on its same side to intersect given circles at D and E.
- Draw a line joining D and E. This line is the required tangent. FG is the other tangent, which can be drawn similarly.
B. Internal Tangents
Draw the given circles with centres P and Q
- Draw a line segment joining P and Q.
- Bisect PQ at O. Draw a circle with OP as diameter to cut the circle at M and R.
- With centre O and radius OM , draw a circle to cut the other circle at S and N.
- Draw line through M and N. This is the required tangent.
- Similarly , draw a line through R and S , which is the other required tangent.
- With centre O , draw the given circle.
- Draw a diameter AB and divide it into five equal parts (same number of parts as the number of sides) and number them as shown.
- With centres A and B and radius AB , draw arcs intersecting each other at M.
- Draw a line joining points M and A_2 intersecting the circle at I. Then AI is the length of the side of the pentagon.
- Starting from I , step – off on the circle, divisions IJ , JK , KL equal to AI.
- Draw segments AI , IJ , JK ,KL and LI thus completing the pentagon.
Let A , B , C , D and E are the given points.
- Draw line segments joining A with B , B with C , C with D and D with E.
- Draw perpendicular bisectors of AB and BC , intersecting each other at P.
- With P as centre and radius PA (or PB or PC) , construct an arc. This arc will pass through points A , B and C.
- Draw a line through P and C.
- Draw the perpendicular bisector of CD intersecting line through P and C at Q.
- With Q as centre , and radius QC (or QD) , construct an arc , passing through points C and D.
- Repeat the process. Please keep in mind that the centre of the arc is at the intersection of the perpendicular bisector and the line joining the previous centre with the last point of the previous arc.
Let BAC be the give right angle.
- With centre A and any convenient radius , draw an arc cutting AB at Q and AC at P
- With the same radius and centres P and Q , draw arcs cutting the arc of step a) at R and S respectively.
- Draw lines joining A with P and Q. AP and AQ trisect the right angle BAC.
Thus ∠BAR = ∠RAS = ∠SAC = 1/3 ∠BAC
Let AB be the given line segment to be divided into unequal parts say 1/6 ,1/5 ,1/ 4 ,1/3 ,1/ 2 .
- Draw a line segment AB of given length.
- Draw perpendiculars AD and BC at points A and B. Complete the rectangle ABCD
- Join diagonals AC and BD intersecting at E.
- Draw perpendicular from E on AB and find intersection point F.
- Now AF = 1/ 2 AB
- Join D and F. The segment FD intersects the diagonal AC at G. Drop perpendicular from G to A. Then AH = 1/3 AB
- Similarly make constructions as given in the figure to obtain 1/ 4 AB , 1/5 AB and 1/6 AB
