Length of Tangent

Any straight line touching a circle only at one point, then that straight line is called a tangent to the circle. If the line is touching at more than one point, it will not be considered a tangent. A tangent has the following properties:

• A tangent at one point only.
• Tangent cannot pass through the circle
• There can be only one tangent at every point on the circumference of the circle.
• From a point outside the circle, only two tangents can be drawn.

Theorem 1: the length of the tangents drawn from an external point to a circle are equal Proof:

Consider the circle with center O. OA and OB are radii of the circle. There are two tangents PA and PB, drawn from the external point P. Tangent is drawn perpendicular to the radius through the point of contact in the circle.

We know, ∠PAO = ∠PBO = 90°

ΔPAO and ΔPBO, ∠PAO = ∠PBO = 90°

PO is common side for both the triangles,

OA = OB [Radii of the circle]

Therefore, by the RHS congruence theorem,

ΔPAO ≅ ΔPBO

⇒PA = PB (Corresponding parts of congruent triangles)

The above proof can also be done using the Pythagoras theorem as follows,

Since,

∠PAO = ∠PBO = 90°

ΔPAO and ΔPBO are right-angled triangles.

PA2 = OP2 – OA2

Since OA = OB,

PA2 = OP2 – OB2 = PB2

This gives, 

PA = PB

This proves that the tangents drawn to a circle from an external point have equal lengths.

Point of observation

Since,
∠APO = ∠BPO,
OP is the angle bisector of ∠APB.

Therefore, the center of the circle is located on the angle bisector made by two tangents from an external point.

Example 1:

A 12 cm tangent is drawn from a distance 8 cm away from its circumference on the circle. Find the radius of the circle. (Hint: Use a length of tangent formula)

Solution:

Length of the tangent = 12 cm
Distance of the external point from the circle = 8 cm

 Distance of the tangent from the centre of the circle = 8 + r

              Radius of the circle = r

              Using Pythagorean theorem,

                                                                          l2 + r2 = d2

                                                                          r2 = d2 - l2

                                                                          r2 = (8 + r)2 - 122

                                                                          r2 = 82 + r2 + 16r - 122

                                                                          r2 = 64 + r2 + 16r - 144

                                                                          16 r = 80

                                                                          r = 80 / 16 = 5 cm

The radius of the circle is 5 cm.

Example 2:

Find the length of the tangent shown below.

Solution:

The above diagram shows that the circle has one tangent and one secant.

PQ = 10 cm and QR = 18 cm,

Therefore, PR = PQ + QR = (10 + 18) cm

= 28 cm.

⇒ SR2 = PR x RQ

⇒ SR2 = 28 x 18

⇒ SR2 = 504 cm

⇒ √SR2 = √504

⇒ SR = 22.4 cm

This implies that the length of the tangent is 22.4 cm.