**Exercise 11.3**

**Q.1: Draw a circle of radius 6 cm. From a point 10 cm away from its center, construct a pair of tangents to the circle and measure their lengths**.

**Solutions: **

Given that:

Construct a circle of radius 6 cm, and let a point P = 10 cm from its centre, construct a pair of tangents to the circle.

Find the length of the tangents.

We follow the following steps to construct the given:

**Steps of construction:**

**1.** First of all, we draw a circle of radius AB = 6 cm.

**2.** Make a point P at a distance of OP = 10 cm, and join OP.

**3.** Draw a right bisector of P, intersecting OP at Q.

**4.** Taking Q as center and radius OQ = PQ, draw a circle to intersect the given circle at T and T`.

**5.** Join PT and P`T` to obtain the required tangents.

Thus, PT and P`T` are the required tangents.

Find the length of the tangents.

As we know that \(OT \perp PT \; and \; \Delta OPT\)

Therefore,

OT = 6 cm and PO = 10 cm.

In \( \Delta OPT\)

\(PT^{2} = OP^{2} – OT^{2}\)

= \(\left ( 10 \right )^{2} – \left ( 6 \right )^{2}\)

= 100 – 36

= 64

PT = 8 cm

Thus, length of tangents = 8 cm.

**Q.2: 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 center. Draw tangents to the circle from these points P and Q.**

**Solutions:**

**Steps of construction:**

**(i)** Draw a line segment PQ of 14 cm.

**(ii)** Take the midpoint O of PQ.

**(iii)** Draw the perpendicular bisectors of PO and OQ which intersects at points R and S.

**(iv)** With center R and radius RP draw a circle.

**(v)** With center S and radius, SQ draw a circle.

**(vi)** With center O and radius 3 cm draw another circle which intersects the previous circles at the points A, B, C, and D.

**(vii)** Join PA, PB, QC, and QD.

So, PA, PB, QC, and QD are the required tangents.

**Q.3: Draw a line segment AB of length 8 cm. Taking A as the center, draw a circle of radius 4 cm 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.**

**Solution. **

**Steps of construction:**

**(i)** Draw a line segment AB of 8 cm.

**(ii)** Draw the perpendicular of AB which intersects it at C.

**(iii)** With the center, C and radius CA draw a circle.

**(iv)** With centers A and B radius 4 cm and 3 cm, draw two circle which intersects the previous at the points P, Q, R and S.

**(v)** Join AR, AS, BP and BQ

So, AR, AS, BP and BQ are the required tangents.

**Q.4: Draw two tangents to a circle of radius 3.5 cm from a point P at a distance of 6.2 cm from its center.**

**Solution:**

**Steps of construction:**

**1.** Draw a circle with O as a center and radius 3.5 cm.

**2.** Mark a point P outside the circle such that OP = 6.2 cm

**3.** Join OP. Draw the perpendicular bisector XY of OP, cutting OP at Q.

**4.** Draw a circle with Q as center and radius PQ( or OQ), to intersect the given circle at the points T and T`.

**5.** Join PT and PT`.

Here, PT and PT` are the required tangents.

**Q.5: Draw a pair of tangents to a circle of radius 4.5 cm, which are inclined to each other at an angle of \(45^{\circ}\).**

**Solution:**

**Steps of Construction:**

**1.** Draw a circle with center O and radius 4.5 cm.

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

**3.** Construct \(\angle BOC = 45^{\circ}\)

**4.** Draw \(AM \perp AB\)

Here, AP and CP are the pairs of tangents to the circle inclined to each other at an angle of \(45^{circ}\)

**Q.6: Draw a right triangle ABC in which AB = 6 cm, BC = 6 cm and \(\angle B = 90^{\circ}\). Draw BD perpendicular from B on AC and draw a circle passing through the points B, C and D. Construct tangents from A to this circle.**

**Solution:**

**Steps of construction:**

**1.** Draw a line segment AB = 6 cm

**2.** At B, draw \(\angle ABX = 90^{\circ}\)

**3.** With B as center and radius 8 cm, draw an arc cutting ray BX at C.

**4.** Join AC. Thus, \(\Delta ABC\)

**5.** From B, draw \(BD \perp AC\)

**6.** Draw the perpendicular bisector of BC, cutting BC at O.

**7.** With O as center and radius OB ( or OC ), draw a circle. This circle passes through B, C and D.

**8.** Thus, this is the required circle.

**9.** Join OA.

**10.** Draw the perpendicular bisector of OA, cutting OA at E.

**11.** With E as a center and radius AE ( or OE ), draw a circle intersecting the circle with center O at B and F.

**12.** Join AF.

Here, AB and AF are the required tangents.