Inscribing And Circumscribing Regular Polygon

Inscribing and Circumscribing Regular Polygon Example 1: Draw an equilateral triangle of side 3 cm. Draw its circumcirlce. Solution:

Steps of construction 1. Draw Triangle ABC with ruler and compasses. 2. Construct the perpendicular bisectors of sides BC and CA which meet at O. 3. Now OC as radius and O as centre, draw a circle. This is the required circle. Example 2: Draw Triangle ABC in which AB = 4 cm, BC = 5 cm and CA = 6 cm. Draw an in-circle of this triangle. Solution:

Steps of construction 1. Draw Triangle ABC with a ruler and compasses. 2. Draw angle bisectors of ∠A and ∠C , which meet at O. 3. Draw OP perpendicular to AC. 4. Now, O as centre and radius equal to OP, draw a circle. This is the required incircle. Example 3: Draw a triangle PQR in which PQ=PR = 5 cm and QR = 4 cm. Inscribe a circle in it. Also draw its circum-circle. Solution

1. Construct Triangle PQR with ruler and compasses. 2. Draw the perpendicular bisectors of sides PQ and PR which meet at O. 3. Now, O as centre and radius equal to OP draw a circle. This is the required circum-circle. 4. Now, draw angle bisectors of ∠Q and ∠R which meet at O ' . From O ' construct a perpendicular on QR. 5. Now O as centre and radius = OT, draw a circle. This is the required in circle.

Example 4: Draw a square of side 4 cm. Draw its circum circle Solution

Steps of construction 1. Draw the circle with ruler and compasses. 2. Draw its diagonals AC and BD to intersect at O. 3. Now, O as centre and radius equal to OA, draw a circle. This is the required circum circle. Example 5: Draw a square of side 5 cm. Inscribe a circle in it. Solution:

Steps of construction 1. Draw the square ABCD with a ruler and compasses. 2. Join AC and BD to intersect at O. 3. From O draw a perpendicular on AB to meet AB at P. 4. Now, O as centre and radius equal to OP, draw a circle. This is the required in circle. Example 6: Inscribe a hexagon in a circle. Solution:

Steps of construction 1. Draw a circle. 2. Divide the circumference of the circle into 6 equal parts by the same radius of the circle. 3. Join AB, BC, CD, DE, EF and FA. Now ABCDEF is the required inscribed hexagon.

Example 7: Inscribe a circle in a regular hexagon. Solution:

Steps of construction 1. Draw the hexagon ABCDEF. 2. Draw the perpendicular bisectors of the adjacent sides FA and FE, which meet at O. 3. With O as centre and radius equal to OP, draw a circle. This is the required in circle. Example 8: Inscribe a regular octagon in a circle. Solution:

Steps of construction 1. Draw the circle. 2. Draw its diameter GC. 3. Draw the perpendicular bisector EA of GC. 4. Draw OD, the angle bisector of ∠EOC and extend it to H. 5. Draw OF, the angle bisector of ∠GOE and extend it to B. 6. Join AB, BC, CD, DE, EF, FG, GH and HA. This is the required regular octagon. Example 9: Draw the in circle of a regular octagon. Solution:

Steps of construction 1. Draw the octagon with a ruler and compasses. 2. Draw a perpendicular from its centre to any side of it. Let it be OP. 3. With OP as radius, draw a circle. This is the required in circle.

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