Length of Tangent

Definition of Tangent:

If a straight line touches a circle at only one point, it is considered as Tangent to the circle.

If the point is touching the circle at more than one point, then it cannot be considered as a tangent. The tangents to any circle have the following properties.

• A tangent touches the circumference of the circle at only one point.

• A tangent is perpendicular to the radius of the circle at the point of contact.

• Tangents cannot be drawn through a point which lies in the interior of the circle.

• From the point outside a circle, only two tangents can be drawn to the circle.

• Every point on the circumference of the circle has one and only one tangent passing through it.

Tangents drawn to circle from external point (Length of tangent Theorem):

From an external point, only two tangents can be drawn to a circle. These two tangents will have the same length. The length of tangent from an external point to the circle can be determined using Pythagora's theorem as the radius of the circle is perpendicular to the tangent. So, the Pythagorean theorem can be used to find the tangent’s length drawn from a point at a known distance away from the center of the circle. Length of tangent to the circle from an external point is given as:

l= $\sqrt{d^{2} - r^{2}}$

The equation is called the length of the tangent formula.

In the above equation,

‘l’ is the length of the tangent

d is the distance between the center of the circle and the external point from which tangent is drawn

‘r’ is the radius of the circle

Length of Tangent Theorem:

Tangents drawn to a circle from an external point are of equal length. This is a very important theorem. It can be proved as shown below.

Length of Tangent Theorem Statement:

Tangents drawn to a circle from an external point are of equal length.

Data:

Consider a circle with the center ‘O’.

Let ‘A’ be the external point at a certain distance away from the center of the circle from which two tangents AB and AC are drawn to the circle at points B and C respectively.

To Prove:

Length of AB = Length of AC

Construction:

Join OB and OC (OB and OC represents the radii of the circle)

Proof:

 Statement Reason In ⊿AOB and ⊿AOCOB = OC Radii of the same circle are equal In ⊿AOB and ⊿AOCOA = OA Side common to both the triangles In ⊿AOB and ⊿AOC∟ABO = ∟ACO Tangents are perpendicular to the radii ⊿AOB ≈ ⊿AOC Side - Angle - Side congruence rule AB = AC Corresponding sides of congruent triangles

Therefore, the length of the tangents drawn to a circle from the same external point are equal.

Length of Tangents Example Problems:

1. A tangent is drawn to a circle of radius 5 cm from a point 8 cm away from the circumference of the circle. Find the length of the tangent.

Solution:

Radius of the circle (r) = 5 cm

Distance of the external point from the circle = 8 cm

Distance of the external point from the center (d) = 8 + r = 8 + 5 = 13 cm

Length of the tangent formula is:

l = $\sqrt{d^{2}-r^{2}}$

l = $\sqrt{13^{2}-5^{2}}$

l = $\sqrt{169-25}$

l = $\sqrt{144}$

l = 12 cm

Length of the tangent = 12 cm

1. A tangent of length 24 cm is drawn to a circle from a distance 18 cm away from its circumference. Find the radius of the circle. (Hint: Use a length of tangent formula)

Solution:

Length of the tangent = 24 cm

Distance of the external point from the circle = 18 cm

Distance of the tangent from the center of the circle = 18 + r

Radius of the circle = r

Using Pythagorean theorem,

l2 + r2 = d2

r2 = d2 - l2

r2 = (18 + r)2 - 242

r2 = 182 + r2 + 36r - 242

r2 = 324 + r2 + 36r - 576

36 r = 252

r = 252 / 36 = 7 cm

The radius of the circle is 7 cm.

1. A circle is inscribed inside a quadrilateral ABCD. Prove that AB + CD = AD + BC

Solution:

Consider a quadrilateral ABCD inside which a circle is inscribed. Let the circle touch the sides of the quadrilateral AB, BC, CD, and DA at the points M, N, O and P respectively as shown in the diagram below.

From the figure,

AM = AP → (1)  (AM and AP are the tangents drawn to the circle from the point A)

BM = BN → (2) (BM and BN are the tangents drawn to the circle from the point B)

CO = CN → (3) (CN and CO are the tangents drawn to the circle from the point C)

DO = DP → (4) (DO and DP are the tangents drawn to the circle from the point D)

Tangents drawn to a circle from an external point are of equal length.

Adding the equations (1), (2), (3) and (4) we get

AM + BM + CO + DO = AP + BN + CN + DP → (5)

From the figure,

AM + BM = AB

CO + DO = CD

BN + CN = BC

Substituting the above values in (5), we get

AB + CD = AD + BC

Fun Facts:

• If a line touches the circle at two different points, then it is called a secant.

• Number of tangents drawn to the circle from a point

• Inside the circle = 0

• On the circle = 1

• Outside the circle = 2