### EXERCISE 11.3

**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:

To construct a circle of radius 6 cm and a pair of tangents to the circle from a point, say X, 10 cm from its centre.

To Find:

The length of the tangents.

Sample Figure:

**Construction:****Step 1.**Draw a circle of radius 6 cm.**Step 2.**Make a point X at a distance of 10 cm from O, and join OX.**Step 3.**Draw a right bisector of X, intersecting OX at an angle of 90^{o}at Q.**Step 4.**Draw a circle to intersect the given circle at T and T`, taking Q as center and radius OQ = XQ.**Step 5.**Join XT and X`T` to get the required tangents.Hence, XT and X`T` are the tangents.

To find the length of the tangent, we know that OTâŠ¥PT and Î”OPT is the right triangle.

Then,

Since, OT = 6 cm and PO = 10 cm.

From Î”OPT,

PT

^{2 }=OP^{2}â€“OT^{2}=(10)^{2}â€“(6)^{2}[Using Pythagoras Theorem]= 100 â€“ 36

= 64

PT = 8 cm

âˆ´, length of tangents = 8 cm.

**Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameters each at a distance of 7 cm from its center. Draw tangents to the circle from these points P and Q.****Solutions:**Given:

To construct a circle of radius 3 cm and a tangents to the circle from the points P and Q, where P and Q are two points on one of the extended diameters of the circle, each at a distance of 7 cm from the center of the circle.

Sample Figure:

**Construction:****Step 1.**First, draw a line segment PQ of 14 cm.**Step 2.**Take the midpoint O of PQ and draw a circle keeping O as the centre of the circle and radius 3cm.**Step 3.**Draw the perpendicular bisectors of PO and OQ which intersects at points R and S, i.e., R and S are the mid points of OP and OQ respectively.**Step 4.**Draw a circle**Step 5.**Draw a circleWith center S and radius SQ.**Step 6.**Here the circle with centre O will be intersecting the other circles with centre R and S at the points A, B, C, and D.**Step 7.**Join PA, PB, QC, and QD.Here, PA, PB, QC, and QD are the tangents required.

**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.****Solutions:**Given:

To draw a line segment AB of length 8 cm and then draw 2 circle of radius 4 cm and 3 cm, taking A and B as the centre of the circles respectively. And later construct tangents to each circle from the center of the other circle.

Sample Figure:

**Construction:****Step 1.**Draw a line segment AB of length 8 cm.**Step 2.**Now, draw the perpendicular bisector of AB which intersects it at C.**Step 3.**Draw a circle with the center, C and radius CA.**Step 4.**Draw two circles with centers A and B of radius 4 cm and 3 cm respectively, such that they intersect the circle with centre C at the points P, Q, R and S.**Step 5.**Join AR, AS, BP and BQHere, AR, AS, BP and BQ are the tangents required.

**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.****Solutions:**Given:

To draw a circle of radius 3.5 cm and two tangents to the circle from a point P at a distance of 6.2 cm from its center.

Sample Figure:

**Construction:****Step 1.**Draw a circle with center O and radius 3.5 cm.**Step 2.**Mark a point P outside the circle such that OP = 6.2 cm**Step 3.**Join OP. and draw the perpendicular bisector of OP say XY, cutting OP at Q.**Step 4.**Draw a circle with center Q and radius PQ or OQ, such that it intersects the given circle at the points T and T`.**Step 5.**Join PT and PT` to obtain the required tangents.Here, PT and PT` are the tangents.

**Draw a pair of tangents to a circle of radius 4.5 cm, which are inclined to each other at an angle of**45^{o}.**Solutions:**To draw a circle or radius 4.5 cm and a pair of tangents to the circle at an inclination of 45

^{o }to each other.Sample figure:

**Construction:****Step 1.**First, draw a circle with center O and radius 4.5 cm.**Step 2.**Draw a diameter of the circle, say AOB.**Step 3.**Construct an âˆ BOC=45^{O}such that, radius OC cuts the circle at C.**Step 4.**Draw AMâŠ¥AB and CNâŠ¥OC. Let AM and CN intersect each other at X.Here, AX and CX are the pairs of tangents to the circle inclined to each other at an angle of 45

^{o}**Draw a right triangle ABC in which AB = 6 cm, BC = 6 cm and**âˆ B=90^{O}**. 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.****Solutions:**Given:

To draw a right triangle ABC in which AB = 6 cm, BC = 6 cm and âˆ B=90O. And then, draw: BD perpendicular from B on AC and a circle passing through the points B, C and D. Later construct tangents from A to this circle.

Sample figure:

**Construction:****Step 1.**Draw a line segment AB of length 6 cm**Step 2.**Draw âˆ ABX=90^{O}at B,.**Step 3.**Draw an arc cutting the ray BX at C, with center B and radius 8 cm,.**Step 4.**Join AC. Here, Î”ABC is the required triangle.**Step 5.**Draw BDâŠ¥AC from B,.**Step 6.**Draw the perpendicular bisector of BC such that it cuts BC at O.**Step 7.**Draw a circle with center O and radius OB or OC. Such that the circle passes through B, C and D. Here, we get the required circle.**Step 9.**Join OA. And draw the perpendicular bisector of OA such that it cuts OA at E.**Step 11.**Draw a circle intersecting the circle with center O at B and F with center E and radius AE or OE.**Step 12.**Now Join AF.Here, AB and AF are the tangents.

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