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103 LearnersLast updated on September 10, 2025
Properties of Tangent
A tangent is a line that touches a curve at a single point without crossing it. Understanding the properties of tangents is crucial in solving geometric problems related to circles. The properties of a tangent include: it is perpendicular to the radius at the point of contact, and tangents from a common external point are equal in length. These properties assist students in analyzing and solving problems involving circles, angles, and lengths. Now let us learn more about the properties of a tangent.
What are the Properties of a Tangent?
The properties of a tangent are straightforward and help students understand and work with this geometric concept. These properties are derived from the principles of geometry. There are several properties of a tangent, and some of them are mentioned below:
- Property 1: Perpendicularity A tangent to a circle is perpendicular to the radius at the point of contact.
- Property 2: Equal Tangents Tangents drawn from a common external point to a circle are equal in length.
- Property 3: Tangent-Secant Theorem If a tangent and a secant are drawn from a point outside the circle, the square of the length of the tangent segment is equal to the product of the lengths of the whole secant segment and its external part.
- Property 4: Angle Between Tangents The angle between two tangents drawn from an external point is supplementary to the angle subtended by the line segment joining the points of contact at the center.
Tips and Tricks for Properties of a Tangent
Students often get confused and make mistakes while learning the properties of tangents. To avoid such confusion, we can follow these tips and tricks:
- Perpendicularity at Point of Contact: Students should remember that a tangent is always perpendicular to the radius at the point where it touches the circle. Verifying this with a diagram can help in understanding.
- Equal Lengths from an External Point: Students should remember that tangents drawn from the same external point to a circle are always equal in length.
- Applying the Tangent-Secant Theorem: Students should practice using the tangent-secant theorem, which relates the lengths of tangents and secants from a common external point.
Confusing a Tangent with a Secant
Students should remember that a tangent only touches the circle at one point, while a secant intersects the circle at two points.
Mistake 1
Misinterpreting the Perpendicularity
Students should know and remember that the tangent is always perpendicular to the radius at the point of contact.
Mistake 2
Misusing the Tangent-Secant Theorem
Students should practice the tangent-secant theorem, ensuring they understand that the square of the tangent segment equals the product of the secant segment's total length and its external part.
Mistake 3
Forgetting the Equal Tangents Rule
Students must remember that tangents from the same external point to a circle are equal in length.
Mistake 4
Solved Examples on the Properties of Tangents
A tangent PA touches a circle at point A. If the radius OA is 5 cm and PA is 12 cm, what is the distance from the center O to the point P?
OP = 13 cm.
Problem 1
Since PA is a tangent, OA is perpendicular to PA. By the Pythagorean theorem in the right triangle OAP, OP² = OA² + PA² = 5² + 12² = 25 + 144 = 169. Thus, OP = √169 = 13 cm.
From an external point, two tangents are drawn to a circle. If one tangent is 9 cm long, what is the length of the other tangent?
Explanation
The other tangent is 9 cm long.
Problem 2
Tangents drawn from the same external point to a circle are equal in length. Therefore, both tangents are 9 cm.
A tangent and a secant are drawn from a point P outside a circle. The tangent segment PT is 4 cm long, and the secant segment PAB cuts the circle at points A and B such that PA = 3 cm and PB = 7 cm. Verify the tangent-secant theorem.
Explanation
The theorem holds true as PT² = PA * PB.
Problem 3
According to the tangent-secant theorem, PT² = PA * PB. Here, PT = 4 cm, PA = 3 cm, PB = 10 cm (PA + AB). PT² = 4² = 16, and PA * PB = 3 * 10 = 30. The values do not satisfy the theorem, which indicates a need to recheck the given lengths.
If a tangent to a circle is perpendicular to a radius at the point of contact, and the radius is 6 cm, what is the angle between the tangent and the radius?
Explanation
The angle is 90 degrees.
Problem 4
By definition, a tangent to a circle is perpendicular to the radius at the point of contact, thus forming a right angle of 90 degrees.
Two tangents are drawn from an external point to a circle, subtending angles of 60 degrees at the center. What is the angle between the tangents?
Explanation
The angle between the tangents is 120 degrees.
A tangent is a line that touches a circle at exactly one point without crossing it.
1.How many tangents can be drawn from an external point to a circle?
2.Are tangents from the same external point equal?
3.How do you find the angle between two tangents from an external point?
4.Can a tangent intersect a circle at more than one point?
Common Mistakes and How to Avoid Them in Properties of Tangents
Students tend to get confused when understanding the properties of tangents, and they often make mistakes while solving problems related to these properties.
Here are some common mistakes students make and ways to avoid them.
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Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
