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.

Construction Procedure:

The instruction to construct a pair of tangents to the given circle follows:

1. Draw a circle with centre O with radius 6 cm.

2. Locate a point P that is 10 cm away from the centre O.

3. Through a line join the points O and P.

4. Now construct perpendicular bisector of the line OP.

5. Let the line PO has M as the mid-point.

6. Now measure the length of MO taking M as the centre.

7. Draw a circle using the length MO as the radius.

8. At point Q and R, the circle drawn with the radius of MO intersects the previous circle.

9. Now join PR and PQ.

10. As a result, the required tangents are PQ and PR.

Justification:

The given problem’s construction can be justified by proving that PQ and PR are tangents to a circle of radius 6cm with centre O.

To prove, join the dotted lines representing OQ and OR.

From the above construction,

In the semi-circle ∠PQO is an angle.

As we know that the angle in a semi-circle is a right angle, therefore

∠PQO = 90°

As a result

⇒ OQ ⊥ PQ

PQ must be a tangent of the circle since OQ is the radius of the circle with a radius of 6 cm. We can now similarly prove that PR is a tangent to the circle.

As a result, the above construction is justified.