# 2.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.

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

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Question

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

(1)Draw two concentric circle CEasy Open in App Solution Verified by Toppr 1 and C 2 with common center

0 and radius 4cm and 6cm

(2) Take a point P on the outer circle C

2 and join OP.

(3) Draw the bisector of OP which bisect OP at M'.

(4) Taking M' as center and OM' as radius draw a dotted circle which

cut the inner circle C

1

at two point M and P.

(5) Join PM and PP'. Thus, PM and PP' are required tangent.

On measuring PM and PP'.

PM=PP ′ =4.4cm Bycalculation: InΔOMP,∠PMO=90 0 PM 2 =OP 2 −OM 2

(bypythagorastheorem)

PM 2 =(6) 2 −(4) 2 =36−16=20 PM 2 =20cm PM= 20 =4.4cm Hence,

thelengthofthetangentis4.4cm

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## Construct a Tangent to a Circle of Radius 4 Cm Form a Point on the Concentric Circle of Radius 6 Cm and Measure Its Length. Also, Verify the Measurement by Actual Calculation.

Construct a Tangent to a Circle of Radius 4 Cm Form a Point on the Concentric Circle of Radius 6 Cm and Measure Its Length. Also, Verify the Measurement by Actual Calculation.

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

Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm ?

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### SOLUTION 1

Steps of Construction

Step 1: Mark a point O on the paper

Step 2: With O as center and radii 4cm and 6cm, 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 Q.

Step 6: Draw a circle with Q as center and radius OQ (or PQ), 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’ 4.5 cm (Approx)

Verification by actual calculation

Join OT to form a right ΔOTP (Radius is perpendicular to the tangent at the point of contact)

In right ΔOTP, OP2=OT2+PT2

(Pythagoras Theorem)

⇒PT=OP2-OT2

⇒PT=62-42=36-16=20 ≈ 4.5cm

(OP = 6 cm and OT = 4cm)

### SOLUTION 2

**Steps of construction:**

1. Draw two concentric circles with centre O and radii 4 cm and 6 cm. Take a point P on the outer circle and then join OP.

2. Draw the perpendicular bisector of OP. Let the bisector intersects OP at M.

3. With M as the centre and OM as the radius, draw a circle. Let it intersect the inner circle at A and B.

4. Join PA and PB. Therefore,

PA¯ and PB¯

are the required tangents.

Concept: Construction of Tangents to a Circle

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Chapter 13: Constructions

Q 10 Q 9

### APPEARS IN

RS Aggarwal Secondary School Class 10 Maths

Chapter 13 Constructions

Q 10

2012-2013 (March) Delhi set 3 (with solutions)

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## Ex 11.2, 2

Ex 11.2, 2 Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation. Steps of construction Draw a circle of radius 4 cm with center O Draw concentric circle of radius 6 cm,

**Check sibling questions**

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

Last updated at July 14, 2020 by Teachoo

**Next**: Ex 11.2, 3 →

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### Transcript

Ex 11.2, 2 Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation. Steps of construction Draw a circle of radius 4 cm with center O Draw concentric circle of radius 6 cm, with center O Mark point P on the larger circle 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 4.47 cm each Finding lengths of PQ and PR Join OQ and OR Since tangent is perpendicular to radius ∠PQO = 90° & ∠PRO = 90° Thus, Δ PQO is a right angled triangle, And, PO = radius of bigger circle = 6 cm and OQ = radius of smaller circle = 4 cm By Pythagoras theorem PO2 = PQ2 + OQ2 62 = PQ2 + 42 36 = PQ2 + 16 PQ2 = 36 – 16 PQ2 = 20 PQ = √20 PQ = 2 × 2.236 PQ = 4.47 cm Similarly, PR = 4.47 cm 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.

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### Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.

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