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Answer :

Steps to construct the required figure:

Step 1: Draw concentric circles C and C’ of radius 4cm and 6cm respectively and mark the center as O.

Step 2: Mark two points as A and P such that O,A and P lie on the same line.

Step 3:Draw perpendicular bisector of OP which intersects OP at O’.

Step 4: By taking O’ as the center, draw a circle of radius OO’ which intersects the circle C at point T and Q.

Step 5: Now join PT and PQ.

Step 6: PT and PQ are our required tangents whose length is 4.5cm approx.

**JUSTIFICATION:**

Join OT and OQ,

We have OT ⊥ PT [Radius ⊥ to tangent ]

In the right angled triangle OTP,

\({OP}^2={OT}^2+{PT}^2 \)

\({(6)}^2={(4)}^2+{(PT)}^2 \)

\({PT}^2=36-16=20 \)

\(PT=\sqrt{20}=2\sqrt{5} cm\)

Similarly \(PQ=2\sqrt{5} cm \)

A pair of tangents can be drawn from an external point outside the circle and these tangents have equal length.

\(\therefore \) PT=PQ

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- Q.1 Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.
- Q.2 Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation.
- Q.3 Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q
- Q.4 Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60°
- Q.5 Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.
- Q. 6 Let ABC be a right triangle in which AB = 6 cm, BC = 8 cm and \(\angle \) B = 90°. BD is the perpendicular from B on AC. The circle through B, C, D is drawn. Construct the tangents from A to this circle.
- Q.7 Draw a circle with the help of a bangle. Take a point outside the circle. Construct the pair of tangents from this point to the circle.

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