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# 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.

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

Click here👆to get an answer to your question ✍️ Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct a pair of tangents to the circle and measure their lengths.

Question

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

Steps of Construction:

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1. A circle with radius 6 cm is drawn taking O as centre

2. Point P is marked at 10 cm away from centre of circle.

3. With the half of compass mark M which is the midpoint of OP.

4. Draw a circle with centre M, taking radius MO or MP which intersects the given circle at Q and R.

5. Now join PQ and PR. These are the tangents of the circle.

We know that, the tangent to a circle is perpendicular to the radius through the point of contact.

∴ In △OPQ, OQ⊥QP and in △OPR, OR⊥PR

Hence, both △OPQ and △OPR are right angle triangles.

Applying Pythagoras theorem to both △s, we get:

OP 2 =OQ 2 +PQ 2 and OP 2 =OR 2 +PR 2

∴ 10 2 =6 2 +PQ 2 and 10 2 =6 2 +PR 2 ⇒PQ 2 =100−36=64 and PR 2 =100−36=64 ∴ PQ=PR=8 cm

Hence, the length of the tangents to a circle of radius 6 cm, from a point 10 cm away from the centre of the circle, is 8 cm.

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## Ex 11.2, 1

Ex 11.2, 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. Steps of construction Draw a circle of radius 6 cm Draw point P, 10 cm away from center 3. Join PO. Make perpendicular bisector of P

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## Ex 11.2, 1 - Chapter 11 Class 10 Constructions (Term 2)

Last updated at July 14, 2020 by Teachoo

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### Transcript

Ex 11.2, 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. Steps of construction Draw a circle of radius 6 cm Draw point P, 10 cm away from center 3. Join PO. Make perpendicular bisector of PO Let M be the midpoint of PO. 4. Taking M as centre and MO as radius, draw a circle. 5. Let it intersect the given circle at points Q and R. 6. Join PQ and PR. ∴ PQ and PR are the required two tangents. After measuring, lengths of tangents PQ and PR are 8 cm each. 4. Taking M as centre and MO as radius, draw a circle. 5. Let it intersect the given circle at points Q and R. 6. Join PQ and PR. ∴ PQ and PR are the required two tangents. After measuring, lengths of tangents PQ and PR are 8 cm each. Justification We need to prove that PQ and PR are the tangents to the circle. Join OQ and OR. Now, ∠PQO is an angle in the semi-circle of the blue circle And we know that, Angle in a semi-circle is a right angle. ∴ ∠PQO = 90° ⇒ OQ ⊥ PQ Since OQ is the radius of the circle, PQ has to be a tangent of the circle. Similarly, PR is a tangent of the circle

Next: Ex 11.2, 2 →

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## 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. Give the justification of the construction.

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. Give the justification of the construction.

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. Give the justification of the construction.

### SOLUTION 1

A pair of tangents to the given circle can be constructed as follows.

Step 1

Taking any point O of the given plane as centre, draw a circle of 6 cm radius. Locate a point P, 10 cm away from O. Join OP.

Step 2

Bisect OP. Let M be the mid-point of PO.

Step 3

Taking M as centre and MO as radius, draw a circle.

Step 4

Let this circle intersect the previous circle at point Q and R.

Step 5

Join PQ and PR. PQ and PR are the required tangents.

The lengths of tangents PQ and PR are 8 cm each.

Justification

The construction can be justified by proving that PQ and PR are the tangents to the circle (whose centre is O and radius is 6 cm). For this, join OQ and OR.

∠PQO is an angle in the semi-circle. We know that angle in a semi-circle is a right angle.

∴ ∠PQO = 90° ⇒ OQ ⊥ PQ

Since OQ is the radius of the circle, PQ has to be a tangent of the circle. Similarly, PR is a tangent of the circle.

### SOLUTION 2

Given that

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

Find the length of tangents.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a circle of radius AB = 6 cm.

Step: II- Make a point P at a distance of OP = 10 cm, and join OP.

Step: III -Draw a right bisector of OP, intersecting OP at Q .

Step: IV- Taking Q as centre and radius OQ = PQ, draw a circle to intersect the given circle at T and T’.

Step: V- Joins PT and PT’ to obtain the require tangents.

Thus, PT and P'T' are the required tangents.

Find the length of tangents.

As we know that OT ⊥ PT and ΔOPT is right triangle.

Therefore,

OT = 6cm and PO = 10cm

In ΔOPT, PT2 = OP2 - OT2 PT2 = 102 - 62 PT2 = 100 - 36 PT2 = 64 PT=64=8

Thus, the length of tangents = 8 cm.

Concept: Construction of Tangents to a Circle

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Chapter 9: Constructions - Exercise 9.3 [Page 17]

Q 1 Q 17 Q 2

### APPEARS IN

RD Sharma Class 10 Maths

Chapter 9 Constructions

Exercise 9.3 | Q 1 | Page 17

NCERT Class 10 Maths

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