Examples of Secant of a Circle
Example 1: Consider a circle with a radius of 5 units. Draw a secant line from a point outside the circle, intersecting the circle at points A and B. If the external part of the secant measures 8 units, find the length of the secant segment within the circle.
Let the length of the secant segment within the circle be x. According to the Intersecting Secants Theorem:
Length of Secant Segment × Length of External Segment = Length of the Other Secant Segment × Length of Its External Segment
x × 8 = x × (x + 8)
⇒8x = x2 + 8x
⇒x2 = 8x
⇒x = 8
∴ Length of the secant segment within the circle is 8 units.
Example 2: In a circle with a diameter of 12 units, a secant is drawn from an external point. If the external segment of the secant is 5 units, find the length of the secant segment within the circle.
Since the diameter is twice the radius, the radius of the circle is (12/2 = 6) units.
(Length of Secant Segment)2 + (Radius)2 = (Diameter)2
(Length of Secant Segment)2 + 62 = 122
(Length of Secant Segment)2 + 36 = 144
(Length of Secant Segment)2 = 108
Length of Secant Segment = √108
Length of Secant Segment = 6√3
So, the length of the secant segment within the circle is 6√3 units.
Example 3: Consider a circle with a radius of 8 cm. A secant intersects the circle such that the external segment of one secant is 5 cm and the entire length of the secant is 12 cm. Find the length of the other external segment.
Given,
- Radius (r) = 8 cm
- Length of the entire secant (AB) = 12 cm
- Length of one external segment (AC) = 5 cm
Using the Secant-Secant Power Theorem:
AC × BC = EC × DC
5 × (5 + BC) = 7 × (7 – BC)
Solve the equation:
⇒25 + 5BC = 49 – 7BC
⇒12BC = 24
⇒BC = 2
So, the length of the other external segment, (BC), is 2 cm.
Example 4: In a circle with a radius of 10 m, a secant is drawn. The length of the entire secant is 16 m, and one of the external segments is 6 m. Determine the length of the other external segment.
Given,
- Radius (r) = 10 m
- Length of the entire secant (AB) = 16 m
- Length of one external segment (AD) = 6 m
Using the Secant-Secant Power Theorem:
AD × BD = CD × ED
⇒6× (6 + BD) = 10 × (10 – BD)
⇒36 + 6BD = 100 – 10BD
⇒16BD = 64
⇒BD = 4
So, the length of the other external segment, (BD), is 4 m.
Secant of a Circle
Secant of a circle is a fundamental concept in geometry that can be described as a straight line intersecting the circle at two distinct points. In this article, we will understand the definition, properties, theorems, and real-world examples surrounding the concept of secants.
In this article, we will learn about the meaning of secant, the formula to calculate the secant of a circle, properties, Intersecting secants, tangent of a circle, theorem of the secant of a circle, the difference between secant, tangent, and chord, and real-life examples of Secant of a Circle.
Table of Content
- What is a Secant of a Circle?
- Formula of Secant of a Circle
- Properties of Secant of a Circle
- Tangent and Secant of a Circle
- Secant of a Circle Theorem
- Examples of Secant of a Circle