Tangent Secant Theorem

The Tangent-Secant Theorem describes the relationship between a tangent and a secant drawn from an external point and has applications in mathematics, engineering, and construction.

Tangent: A tangent is a straight line that touches a circle at exactly one point. The point where it touches is called the point of tangency. The tangent is always perpendicular to the radius at this point.

Secant: A secant is a straight line that intersects a circle at two points, passing through its interior.

Theorem Statement

If a tangent and a secant are drawn from a point outside a circle, then the square of the length of the tangent is equal to the product of the whole length of the secant and the external part of the secant.

👁 Tangent Secant Theorem Statement

In the above figure O is the center of the circle, AB be the tangent of the circle from the external point A and ACD be the secant of the circle where C and D are the points on the circle.

According to the Tangent-Secant Theorem

AB² = AD × AC

Proof

Consider the figure below, where O is the center of the circle ACD is secant of the circle and AB be the tangent on the circle. A line OP is drawn perpendicular to CD. Join OC, OA and OB.

👁 Tangent-and-Secant-2

Now, since OP ⟂ CD

CP = PD ---(1)

[Perpendicular drawn from the center of the circle on the chord bisects the chord]

AC × AD = (AP - CP) (AP + PD)
⇒ AC × AD = (AP - CP) (AP + CP) [From 1]
⇒ AC × AD = AP² - CP²

In △ OAP

OA² = OP² + AP²
⇒ AP² = OA² - OP²
⇒ AC × AD = AP² - CP²
⇒ AC × AD = OA² - OP² - CP²
⇒ AC × AD = OA² - (OP² + CP²)

In △ OCP

OC² = OP² + CP²
⇒ CP² = OC² - OP²
⇒ AC × AD = OA² - (OP² + CP²)
⇒ AC × AD = OA² - (OP² + OC² - OP²)
⇒ AC × AD = OA² - OC²

Since OC = OB

Thus, AC × AD = OA² - OB²

In △ OAB

OA² = OB² + AB²
⇒ AB² = OA² - OB²
⇒ AC × AD = OA² - OB²
⇒ AC × AD = AB²

Hence proved

Applications of Tangent Secant Theorem

• Used in construction of buildings and bridges.
• Helpful in designing arches, domes, and curved structures.
• Used in monuments and statues with circular designs.
• Helps in solving geometry problems related to circles.

Related Articles

• Pythagoras Theorem
• Mid Point Theorem
• Circle.

Solved Problems on Tangent Secant Theorem

Example 1: Find the value of x.

👁 Tangent Secant Theorem: Solved Example 01

Solution:

Total length of the secant = 6 + 10 = 16

By the tangent secant theorem

AB² = AC × AD

⇒ x² = 6 × 16

⇒ x² = 96

⇒ x = 4√6

Example 2: Find the total length of the secant.

Solution:

Using Tangent–Secant Theorem:
AB² = AC × AD

Given: AB = 10, AC = 2, CD = x ⇒ AD = 2 + x

10² = 2(2 + x)
100 = 4 + 2x
2x = 96
x = 48

Total length of secant AD = 2 + 48 = 50.

Example 3: Find the length of the tangent.

Solution:

By the tangent secant theorem

AB² = AC × AD

⇒ z² = 8 × 15

⇒ z² = 120

⇒ z = 2√30

Example 4: Find the value of x.

Solution:

By the tangent secant theorem

AB² = AC × AD

⇒ 12² = x × (x + 5)

⇒ 144 = x² + 5x

⇒ x² + 5x - 144= 0

⇒ x = 9.75 or x = -14.75 (length cannot be negative)

Thus, x = 9.75

Example 5: Find the value of x and y.

Solution:

By the tangent secant theorem

AB² = AC × AD

⇒ 9² = x × 12

⇒ x = 81 / 12

⇒ x = 6.75

From the above figure

x + y = 12

⇒ y = 12 - x

⇒ y = 12 - 6.75

⇒ y = 5.25

Practice Problems on Tangent Secant Theorem

Problem 1: In a circle with a radius of 5 cm, point P is located 13 cm away from the center O. A tangent is drawn from point P to the circle, and it touches the circle at point T. Calculate the length of PT.

Problem 2: In a circle with a radius of 8 cm, a secant is drawn from an external point P. The external portion of the secant is 12 cm, and the entire secant is 16 cm long. Find the length of the tangent segment from point P to the circle.

Problem 3: In a circle with a radius of 6 cm, a secant line is drawn from an external point P such that the tangent segment formed from P to the circle is 8 cm long. Find the length of the entire secant.

Problem 4: In a circle with a radius of 10 cm, a tangent is drawn from an external point P. If the tangent segment PT is 6 cm long, find the length of the secant from P to the circle.

Problem 5: In a circle with a radius of 7 cm, a secant is drawn from an external point P such that the external portion of the secant is 15 cm long, and the entire secant is 20 cm long. Calculate the length of the tangent segment from point P to the circle.

Problem 6: In a circle with a radius of 12 cm, point P is located 19 cm away from the center O. A tangent is drawn from point P to the circle, and it touches the circle at point T. Calculate the length of PT.