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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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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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Construction Procedure:

For the given circle, the tangent can be drawn as follows.

1. Draw a circle of 4 cm radius with centre “O”.

2. Again, take O as centre draw a circle of radius 6 cm.

3. Locate a point P on this circle

4. Join the points O and P through lines such that it becomes OP.

5. Draw the perpendicular bisector to the line OP

6. Let M be the mid-point of PO.

7. Draw a circle with M as its centre and MO as its radius

8. The circle drawn with the radius OM, intersect the given circle at the points Q and R.

9. Join PQ and PR.

10. PQ and PR are the required tangents.

It can be calculated manually as follows

In ∆PQO,

Since PQ is a tangent,

∠PQO = 90°. PO = 6cm and QO = 4 cm

Applying Pythagoras theorem in ∆PQO, we obtain

PQ2+QO2 = PQ2

PQ2+(4)2 = (6)2

PQ2 +16 =36

PQ2 = 36−16

PQ2 = 20

PQ = 2√5

PQ = 4.47 cm

Therefore, the tangent length PQ = 4.47

To prove this, join OQ and OR represented in dotted lines.

From the construction,

∠PQO is an angle in the semi-circle.

We know that angle in a semi-circle is a right angle, so it becomes,

∴ ∠PQO = 90°

Such that

⇒ OQ ⊥ PQ

Since OQ is the radius of the circle with radius 4 cm, PQ must be a tangent of the circle. Similarly, we can prove that PR is a tangent of the circle.

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