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.
What is a Secant of a Circle?
A secant of a circle is a straight line that intersects the circle at two distinct points. When this line crosses a circle, it enters the interior of the circle, creating two points of intersection on the circle itself. Essentially, a secant line connects two points on the circle's circumference by passing through its interior. It's important to note that a straight line can intersect any given circle at a maximum of two different points, and when it does, it is referred to as a secant line to that circle.
Secant of Circle Definition
A secant of a circle is a straight line that intersects the circle at two distinct points. This line passes through the circle, creating two points of intersection on its boundary.
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The formula for the length of a secant line (s) in a circle with radius (r) and the central angle (θ) is given by:
s = 2 × r × sin (θ/2)
where,
- s represents Length of Secant Line
- r is Radius of Circle
- θ is Measure of Central Angle formed by Secant in Radians
This formula helps in calculating the length of a secant line in a circle based on the radius of the circle and the measure of the central angle it subtends.
Properties of Secant of a Circle
A secant of a circle is a straight line that crosses the circle at two points, it has a few features given below:
- Two Intersection Points: The secant intersects the circle at two distinct points. These points are on the circle.
- Interior Crossing: The secant goes inside the circle after intersecting it. It doesn't stay on the circle's boundary.
- Maximum Intersection: A secant line can intersect a circle at a maximum of two different points. It cannot cross the circle at more than two points.
Intersecting Secants
When two secant lines cross or intersect inside a circle, they are referred to as intersecting secants. In simple terms, intersecting secants are two straight lines that both cut through the circle and meet or cross each other within the circle's boundaries. This interaction creates points of intersection and various geometric relationships within the circle, including the formation of chords, central angles, and the division of the circle into segments.
- Multiple Intersection Points: Intersecting secants create four distinct points of intersection within the circle. These points mark the locations where the two secant lines cross each other.
- Segment Formation: The circle is divided into segments by the intersecting secants. These segments include the area between the points of intersection, as well as the regions outside both secants.
- Chord Creation: Each secant produces a chord within the circle, connecting two of the points of intersection. Therefore, when two secants intersect, they collectively generate two chords, each associated with one of the intersecting secants.
- Central Angle Formation: The central angle formed by the intersection of two secants is an angle whose vertex is at the center of the circle. This angle is determined by the lines connecting the center of the circle to the points of intersection.
Tangent and Secant of a Circle
A tangent is a line that connects with the circle at just one point, while a secant is a line that intersects the circle at two points. It is a specific type of secant which occurs when the two endpoints of the secant's chord come together at a single point.
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Tangent of circle cuts the circle at only one point.
| Secant of circle cuts the circle at two points.
|
Tangent never passes through circle.
| Secant passes through the circle.
|
Tangent never intersects two points.
| Secant intersects with two points.
|
Tangent is perpendicular to the radius of the circle.
| Secant of the circle may or may not be perpendicular to the circle.
|
Secant of a Circle Theorem
Various theorem related to Secant of a Circle Theorem are added below,
I. Tangent Secant Theorem
Tangent-Secant Theorem states that when you draw a tangent segment and a secant segment from an external point to a circle, the square of the length of the tangent segment is equal to the product of the length of the secant segment and its outer portion.
In a formula: (AB)2 = AC × AD
Learn, Tangent Secant Theorem
II. Intersecting Secants Theorem
The Intersecting Secants Theorem states that if two secant segments are drawn from a point outside a circle, then the product of the length of one secant segment and its outer portion is equal to the product of the length of the other secant segment and its outer portion.
In a formula: MN × MO = MP × MQ
This theorem is applicable when you have two intersecting secants, such as MO and MQ in the provided figure, and it establishes a relationship between the lengths of these secant segments and their external portions.
III. Secant and Angle Measures
When two secant lines intersect inside a circle, the measure of each angle formed is half the sum of the measures of the arcs intercepted by those secants. In simpler terms, if you have two secant lines like AD and BC intersecting inside the circle at point O, then the size of the angle AOB is equal to half the sum of the lengths of the arcs AB and CD.
m∠AOB = 1/2(AB + CD)
Whereas, when two secant lines intersect outside the circle, the measure of the angle formed by these lines is half the positive difference between the measures of the intercepted arcs. For instance, in the circle with intersecting secant lines AC and AE outside the circle at point A, the angle CAE is equal to half the positive difference between the lengths of the arcs CE and BD.
m∠CAE = 1/2(CE - BD)
Difference Between Chord and Secant
Though in a Circle Chord and a Secant look like a straight like, but they are different from each other.
Chord of Circle
When a secant line intersects a circle at two points, it forms a chord between those two points. A chord is a line segment with its endpoints on the circle's circumference. In simpler terms, a chord is like a piece of the circle's edge.
Secant of Circle
A secant is a straight line that cuts through a circle, hitting it at two different points. In the context of a circle, when a secant is drawn, it creates a chord at the points where it intersects the circle. So, a secant is responsible for defining and establishing the chord in a circle by connecting two points on the circle's boundary.
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| Line segment with endpoints on the circle's circumference. | Straight line intersecting the circle at two points.
|
Formed by connecting two points on the circle.
| Created by drawing a straight line that crosses the circle at two distinct points.
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Limited to the distance between its endpoints on the circle.
| Can extend beyond the circle, comprising the segment between its two intersection points and the parts outside the circle.
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Always lies entirely within the circle.
| Partly lies inside and partly outside the circle.
|
If the chord passes through the center of the circle, it becomes a diameter.
| When the secant passes through the center, it also becomes a diameter.
|
A chord has only two intersection points on the circle.
| A secant has two intersection points on the circle, and it includes the chord between these points.
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Real-Life Examples of Secant of a Circle
1. Car Turning at a Roundabout: Imagine a car moving along the circular path of a roundabout. The path the car takes can be compared to a secant of the circular roundabout as it intersects the circle at two points.
2. Flashlight Illuminating a Globe: Picture a flashlight shining on a globe. The beam of light that touches the globe at two opposite points represents a secant as it intersects the spherical surface.
3. Telephone Cable on a Utility Pole: Consider a telephone cable running from the top of a utility pole to the ground. The cable forms a secant of the circular cross-section of the pole, intersecting it at two points.
4. Airplane Flight Path: When an airplane takes off or lands, its flight path may resemble a secant as it intersects the circular shape of the Earth's surface at the departure and arrival points.
5. Ferris Wheel Movement: Think about a Ferris wheel in an amusement park. The support structure of the wheel forms a secant as it intersects the circular shape of the wheel, providing stability and support.
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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.
Practice Questions of Secant of a Circle
Q1. In a circle with a radius of 7 units, a secant line is drawn from a point outside the circle. If the external part of the secant measures 10 units, find the length of the secant segment within the circle.
Q2. A circle has a diameter of 14 units. If a secant is drawn from an external point, and the length of the external segment is 8 units, calculate the length of the secant segment within the circle.
Q3. Consider a circle with a radius of 9 units. If a secant line intersects the circle at two points, and the length of the secant segment within the circle is 12 units, find the length of the external segment.
Q4. In a circle with a diameter of 20 units, a secant is drawn from an external point. If the external segment of the secant is 15 units, calculate the length of the secant segment within the circle.
Q5. A circle has a radius of 5 units. If a secant line is drawn from a point outside the circle, and the length of the secant segment within the circle is 3 units, find the length of the external segment.
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