# 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.

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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.

Steps of Construction:Medium Open in App Solution Verified by Toppr

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

OQ=OR=radius=6 cm and OP=10 cm

∴ 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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### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

## 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.

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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.

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### 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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