**Q.1: Draw a circle of radius 3cm. From a point P, 7cm away from the center of the circle draw two tangents to the circle. Also, measure the lengths of the tangents.**

**Steps of Construction:**

**Step 1: **Draw a circle of radius 5 cm with center O.

**Step 2: **A point P at a distance of 8cm from O is taken.

**Step 3: **A right bisector of OP meeting OP at M is drawn.

**Step 4: **With center M radius OM a circle is drawn intersecting the previous circle at T_{1 }and T_{2}

**Step 5: **Join PT_{1} and PT_{2}

PT_{1} and PT_{2} and the required tangents, Measuring PT_{1} and PT_{2}

We find, PT_{1} = PT_{2} = 6.2 cm

**Q.2: Draw two tangents to a circle of radius 3.5 cm from a point P at a distance of 6.2cm from its center.**

**Steps of construction: **

**Step 1: **Two concentric circles with centre O and radii 4cm and 6 cm are drawn.

**Step 2: **A point P is taken on outer circle and O, P are joined.

**Step 3: **A right bisector of OP is drawn bisecting OP at M.

**Step 4: **With centre M and radius OM a circle is drawn cutting the inner circle at T_{1} and T_{2}

**Step 5: **Join PT_{1 }and PT_{1}

PT_{1} and PT_{2} are the required tangents. Further PT_{1} = PT_{2} = 4.8 cm.

**Q.3:Draw a circle of radius 3.5cm. Take two points A and B on one of its extended diameter, each at a distance of 5cm from its centre. Draw tangents to the circle from each of these points A and B.**

**Steps of construction: **

**Step 1: **Draw a circle with center O and radius 3.5 cm

**Step 2: **the diameter P_{1}P_{2} is extended to the points A and B such that AO = OB = 7 cm.

**Step 3: **With center P_{1} and radius 3.5 cm draw a circle cutting the first circle at T_{1} and T_{2}

**Step 4: **Join AT_{1} and AT_{2}

**Step 5: **With center P_{2} and radius 3.5 cm draw another circle cutting the first circle at T_{3} and T_{4}

**Step 6: **Join BT_{3} and BT_{4}. Thus AT_{1}, AT_{2} and BT_{3}, BT_{4} are the required tangents to the given circle from A and B.

**Q.4:Draw a circle with centre O and radius 4cm. Draw any diameter AB of this circle. Construct tangents to the circle at each of the two end points of the diameter AB.**

**Steps of Construction:**

**Step 1: **A circle of radius 4.2 cm at center O is drawn.

**Step 2: **A diameter AB is drawn.

**Step 3: **With OB as the base, an angle BOC of 45^{o} is drawn.

**Step 4: **At A, a line perpendicular to OA is drawn.

**Step 5: **AT C, a line perpendicular to OC is drawn.

**Step 6: **These lines intersect with each other at P.

PA and PC are the required tangents.

**Q.5:Draw a circle with the help of a bangle. Take any point P outside the circle. Construct the pair of tangents from the point P to the circle. **

**Steps of Construction: **

**Step 1: **A line segment AB = 8, 5 cm is drawn.

**Step 2: **Draw a right bisector of AB which meets AB at M.

**Step 3: **With M as center AM as radius a circle is drawn intersecting the given circles at T_{1}, T_{2}, T_{3} and T_{4}

**Step 4: **Join AT_{3}, AT_{4} and BT_{1} and BT_{2}

Thus AT_{3}, AT_{4}, BT_{1}, BT_{2 }are the required tangents.

**Q.6: Draw a line segment AB of length 8cm. Taking A as the centre, draw a circle of radius 4cm and taking B as the center, draw another circle of radius 3 cm. Construct tangents to each circle from the center of the other circle.**

**Step 1: **Draw a line segment AB = 7cm

**Step 2: **Taking A as center and radius 3 cm, a circle is drawn.

**Step 3: **With center B and radius 2.5 cm, another circle is drawn.

**Step 4: **With center A and radius more than ½ AB, arcs are drawn of both sides of AB.

**Step 5: **With center B and the same radius [as in step 4] arcs are drawn on both sides of AB intersecting previous arcs at P and Q.

**Step 6: **Join PQ which meets AB at M.

**Step 7: **With center M and radius AM, a circle is drawn which intersects circle with center A at T_{1} and T_{2} and the circle with the center B at T_{3} and T_{4}.

**Step 8: **Join AT_{3}, AT_{4}, BT_{1} and BT_{2}

Thus AT_{3}, AT_{4}, BT_{1} and BT_{2} are the required tangents.

**Q.7:Draw a circle of radius 4.32cm. Draw a pair of tangents to this circle inclined to each other at an angle of 45°.**

**Steps of construction:**

**Step 1: **A circle of radius 3 cm with center O is drawn.

**Step 2: **A radius OC is drawn making an angle of 60^{o} with the diameter AB.

**Step 3: **At C, \(\angle\)^{o} is drawn.

CP is required tangent.

**Q.8:Write the steps of construction for drawing a pair of tangents to a circle of radius 3cm, which are inclined to each other at an angle of 60°.**

Steps of Construction:

**Step 1: **Draw a circle with center O and radius 3 cm.

**Step 2: **Draw any diameter AOB of the circle.

**Step 3: **Construct ∠BOC = 60° such that radius OC cuts the circle at C.

**Step 4: **Draw AM perpendicular to AB and CN perpendicular to OC. Suppose AM and CN intersect each other at P.

Here, AP and CP are the pair of tangents to the circle inclined to each other at an angle of 60°.

**Q.9:Draw a circle of radius 3cm. Draw a tangent to the circle making an angle of 30° with a line passing through the center.**

Steps Of construction:

**Step 1: **Draw a circle with center O and radius 3 cm.

**Step 2: **Draw radius OA and produce it to B.

**Step 3: **Make ∠AOP = 60°.

**Step 4: **Draw PQ perpendicular to OP, meeting OB at Q.

**Step 5: **Then, PQ is the desired tangent, such that ∠OQP = 30°.

**Q.10:Construct a tangent to a circle of radius 4cm from a point on the concentric circle of radius 6cm and measure its length. Also, verify the measurement by actual calculation.**

Steps of Construction:

**Step 1: **Mark a point O on the paper.

**Step 2: **With 0 as center and radii 4 cm and 6 cm, draw two concentric circles.

**Step 3: **Mark a point P on the outer circle.

**Step 4: **Join OP.

**Step 5: **Draw the perpendicular bisector XY of OP, cutting OP at O.

**Step 6: **Draw a circle with O as center and radius OQ (or P0), to intersect the inner circle in points, T and T’.

**Step 7: **Join PT and PT’.

Here, PT and PT’ are the required tangents.

PT = PT’ = 4.5cm (Approx)