Case I: When the centre of the circle is known.
Construction 3: Draw two tangents to a circle (whose diameter is 6 cm) from a point P at a distance 9 cm from its centre. Also measure the length of two tangents.
Steps of Construction:
1. Draw a circle with centre O and whose radius is 3 cm.
2. Take a point P at a distance 9 cm from the centre O.
3. Join O with P i.e., OP = 9 cm.
4. Bisect OP at C or draw the right bisector of OP.
5. With C as centre and radius PC or OC describe another circle cutting the given circle at T and T.
6. Join PT and PT.
7. PT and PT are required tangents drawn from a point P to the given circle.
Case II: When the centre of the circle is not known.
Construction 4: Draw a circle of radius 2 cm and construct tangents to it from an external point P without using the centre.
Steps of Construction:
1. Draw a circle with radius 2 cm.
2. Through the external point P draw a secant PBA meeting the circle at A and B.
3. Bisect AP at O or draw the right bisector of AP.
4. Draw a semi-circle described on AP as diameter.
5. Through B draw BC ^ AP meeting the semi-circle at C.
6. With centre P and radius PC draw an arc cutting the given circle at T and T.
7. Join PT and PT' which are the required tangents.
Construction 5: Construct a circle of radius 3 cm and take a point P at a distance of 7.5 cm from its centre. From it, construct tangents to the circle without using its centre.
Steps of Construction:
1. Draw a circle with centre O with radius 3 cm and P is a point at a distance 7.5 cm from O.
2. Draw a secant through P intersect the circle at A and B.
3. Produce AP to a point C such that PA = PC.
4. With BC as diameter, draw a semi circle.
5. At P, construct a perpendicular to BC to intersect the semi-circle at D.
6. With P as centre and PD as radius, draw an arc to intersect the given circle at T and T.
7. Join PT and PT. Then, PT and PT are the two required tangents from the point P to the given circle.
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