Question

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

Answer 1

Step 1 Construct a circle from given conditions

• Let us draw a circle with radius $=6cm$ with centre O.
• Take a point P, which is $10cm$ away from O.
• Join the points O and P through a line.

Step 2: Construct the two tangents from point P to the circle drawn in step 1

• Draw the perpendicular bisector of line PO.

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

• Take M as a centre and measure the length of MO then the length MO is taken as radius and draw a circle.
• The circle drawn with the radius of MO, intersect the previous circle at point Q and R.
• Join PQ and PR .
• so PQ and PR are the required tangents.

Step 3 Justification

We have to prove that PQ and PR are the tangents to the circle of radius 6cm with centre O.

Let us join OQ and OR represented in dotted lines.

From the construction,

$\angle PQO$ is an angle in the semi-circle.

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

$\therefore \angle PQO=90°$

Such that

$\Rightarrow OQ\perp PQ$

Since OQ is the radius of the circle with a radius 6 cm, PQ must be a tangent of the circle.

Similarly, we can prove that PR is a tangent of the circle.

Hence, the circle is constructed with the help of given condition and justification is also provided