Question

Draw a circle of radius$3cm$. Take two points P and Q on one of its extended diameter each at a distance of$7cm$ from its centre. Draw tangents to the circle from these two points P and Q. Give the justification of the construction.

Answer 1

To construct a tangent from given conditions

1. Let us draw a circle with a radius of$3cm$ with centre “O”.
2. Draw a diameter of a circle and it extends $7cm$ from the centre and marks it as P and Q.
3. Draw the perpendicular bisector of the line PO and mark the midpoint as M.
4. Draw a circle with M as centre and MO as the radius.
5. Now join the points PA and PB in which the circle with radius MO intersects the circle of circle $3cm$.
6. Now PA and PB are the required tangents.
7. Similarly, from point Q, we can draw the tangents.
8. From that, QC and QD are the required tangents.

Justification:

We have to prove that PA and PB are the tangents to the circle of radius $3cm$ with centre O.

To prove this, join OA and OB.

From the construction,

∠PAO is an angle in the semi-circle.

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

$\angle PAO=90°$

Such that

$\Rightarrow$$OA\perp PA$

Since OA is the radius of the circle with radius of $3cm$, PA must be a tangent of the circle.

Similarly, we can prove that PB, QC and QD are the tangents of the circle.