Question 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.

**Solution:** Draw a circle with radius 6 cm and centre O.

- Draw a line segment OP = 10 cm
- Make perpendicular bisector of OP which intersects OP at point O’.
- Take O’P as radius and draw another circle.
- From point P, draw tangents to points of intersection between the two circles.
- PQ = PR = 8 cm.

**Justification:** Radius, tangent and distance between centre and external point (from which tangent is drawn) make a right triangle. Using Pythagoras theorem, we have;

`PQ^2 = OP^2 – OQ^2`

`= 10^2 – 6^2`

`= 100 – 36 = 64`

Or, `PQ = 8 cm`

Question 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.

**Solution:** Draw two concentric circles with radii 4 cm and 6 cm.

- Draw the radius OP of bigger circle.
- Make a perpendicular bisector of OP which intersects OP at point O’.
- Taking O’P as radius, draw another circle.
- From point P, draw tangents PQ and PR; as shown in figure.

**Justification:** Using Pythagoras theorem, we have;

`PQ^2 = OP^2 – OQ^2`

`= 6^2 – 4^2`

`= 36 – 16 = 20`

Or, `PQ = 2sqrt5 cm`

Question 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.

**Solution:** Draw a circle with radius 3 cm.

- Extend the diameter to A and B on either side.
- Draw perpendicular bisectors of OA and OB.
- Perpendicular bisector of OA intersects it at point O’.
- Perpendicular bisector of OB intersects it at point O”.
- Taking O’A as radius, draw the second circle.
- Taking O”B as radius, draw the third circle.
- From point A, draw tangents AP and AQ.
- From point B, draw tangents BR and BS.

Question 4: Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60^{o}.

**Solution:** Draw a circle with radius 5 cm.

- Draw a line AP perpendicular to radius OA.
- Draw OB at 120
^{o}from OA. (Because angle at centre is double the angle between two tangents). - Join A and B to P; to get two tangents.
- Here, ∠APB = 60
^{o}

Question 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.

**Solution:** Draw a line segment, AB = 8 cm.

- Taking A as centre, draw a circle with radius 4 cm.
- Taking B as centre, draw another circle with radius 3 cm.
- Draw perpendicular bisector of AB.
- Taking midpoint of AB as centre and AB as diameter, draw the third circle.
- From point A, draw tangents AR and AS.
- From point B, draw tangents BP and BQ.

Question 6: Let ABC be a right triangle in which AB = 6 cm, BC = 8 cm and ∠B = 90^{o}. BD is the perpendicular from B on AC. The circle through B, C, D is drawn. Construct the tangent from A to this circle.

**Solution:** Draw a line segment AB = 6 cm.

- Make a right angle at point B and draw BC = 8 cm.
- Draw a perpendicular BD to AC.
- We know that in a right triangle, hypotenuse is the diameter of circumcircle.
- Taking BC as diameter, draw a circle which passes through points B, C and D.
- Join A to O and taking AO as diameter, draw second circle.
- From point A, draw tangents AB and AP.

Question 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.

**Solution:** This question can be solved as the first question.

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