**7) Draw a circle with radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.**

**Solution:**

**Procedure for construction:**

- Draw a line segment of length AB = 10 cm. Bisect AB by constructing a perpendicular bisector of AB. Let M be the mid-point of AB.
- With M as centre and AM as radius, draw a circle. Let it intersect the given circle at the points P and Q.
- Join PB and QB. Thus, PB and QB are the required two tangents.

**Justification:** Join AP. Here ∠APB is an angle in the semi-circle. Therefore, ∠APB = 90°. Since AP is a radius of a circle, PB has to be a tangent to a circle. Similarly, QB is also a tangent to a circle.

In a Right ∆APB, AB^{2} = AP^{2} + PB^{2} (By using Pythagoras Theorem)

PB^{2} = AB^{2} – AP^{2} = 10^{2} — 6^{2} = 100 – 36 = 64

PB = 8 cm.

**8) 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.**

**Solution:**

**Procedure for construction:**

- Draw a line segment of length OA = 4 cm. With O as centre and OA as radius, draw a circle.
- With O as centre draw a concentric circle of radius 6 cm(0B).
- Let C be any point on the circle of radius 6 cm, join OC.
- Bisect OC such that M is the mid point of OC.
- With M as centre and OM as radius, draw a circle. Let it intersect the given circle of radius 4 cm at the points P and Q.
- Join CP and CQ. Thus, CP and CQ are the required two tangents.

**Justification:**

Join OP. Here ∠OPC is an angle in the semi-circle. Therefore, ∠OPC = 90°. Since OP is a radius of a circle, CP has to be a tangent to a circle. Similarly, CQ is also a tangent to a circle.

In ∆COP, ∠P = 90°

=

**9) Draw a circle with radius 3 cm. On one of its extended diameter, take two points P and Q each at a distance of 7 cm from its centre. From two points P and Q, draw tangents to the circle.**

**Solution:**

**Given:**

Two points P and Q on the diameter of a circle with radius 3 cm OP = OQ = 7 cm.

**Aim**:

To construct the tangents to the circle from the given points P and Q.

**Procedure for construction:**

- Draw a circle with radius 3 cm with centreO.
- Extend its diameter both the sides and cut OP = OQ = 7 cm.
- Bisect OP and OQ.Let mid-points of OP and OQ be M and N.
- With M as centre and OM as radius, draw a circle. Let it intersect (0, 3) at two points A and B. Again taking N as centre ON as radius draw a circle to intersect circle(0, 3) at points C and D.
- Join PA, PB, QC and QD. These are the required tangents from P and Q to circle (0, 3).

**10) Draw a pair of tangents to a circle which is of radius 5 cm, such that they are inclined to each other at an angle of 60°.**

**Solution:**

**To determine**: To draw tangents at the ends of two radius which are inclined to each other at 120°

**Procedure for construction : **

- Keeping O as centre, draw a circle of radius 5 cm.
- Take a point Q on the circle and join it to O.
- From OQ, Draw∠QOR = 120°.
- Take an external point P.
- Join PR and PQ perpendicular to OR and OQ respectively intersecting at P.

The required tangents are RP and QP.