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

#### Solution 1

The tangent can be constructed on the given circle as follows.

**Step 1**

Taking any point O on the given plane as centre, draw a circle of 3 cm radius.

**Step 2**

Take one of its diameters, PQ, and extend it on both sides. Locate two points on this diameter such that OR = OS = 7 cm

**Step 3**

Bisect OR and OS. Let T and U be the mid-points of OR and OS respectively.

**Step 4**

Taking T and U as its centre and with TO and UO as radius, draw two circles. These two circles will intersect the circle at point V, W, X, Y respectively. Join RV, RW, SX, and SY. These are the required tangents.

**Justification**

The construction can be justified by proving that RV, RW, SY, and SX are the tangents to the circle (whose centre is O and radius is 3 cm). For this, join OV, OW, OX, and OY.

∠RVO is an angle in the semi-circle. We know that angle in a semi-circle is a right angle.

∴ ∠RVO = 90°

⇒ OV ⊥ RV

Since OV is the radius of the circle, RV has to be a tangent to the circle.

Similarly, it can be shown that RW, SX, and SY are the tangents of the circle.

#### Solution 2

Given that

Construct a circle of radius 3 cm, and let a point P and Q extended diameter each at distance of 7cm from its centre. Construct the pair of tangents to the circle from these two points P and Q.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a circle of radius = 3 cm.

Step: II- Make a line CD = diameter = 6 cm.

Step: III-Extend the line CD in such a way that point CP = DQ = 7 cm

Step: IV- CP at a distance of OP = 7 + 3 = 10 cm, and join OP draw a right bisector of OP, intersecting OP at R.

Step V:- Similarly, DQ at a distance of OQ = 7 + 3 = 10 cm, and join OQ draw a right bisector of OQ, intersecting OQ at S.

Step VI: Taking R and S as centre and radius OS = OR, draw a circle to intersect the given circle at T and T’

B and B ’respectively.

Step: VII- Joins PT and PT’ as well as QB and QB’ to obtain the require tangents.

Thus, PT and P'T' , QB and QB' are the required tangents.