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102 LearnersLast updated on September 10, 2025
Properties of Tangents to a Circle
Tangents to a circle have several unique properties that are essential in solving geometric problems related to circles. These properties help students analyze and solve problems involving tangents, angles, and distances. The properties include: a tangent to a circle is perpendicular to the radius at the point of tangency, and tangents drawn from a common external point are equal in length. Understanding these properties assists students in solving problems related to circles effectively. Now let us learn more about the properties of tangents to a circle.
What are the Properties of Tangents to a Circle?
The properties of tangents to a circle are fundamental and help students understand and work with this geometric concept. These properties are derived from the principles of geometry. There are several properties of tangents to a circle, and some of them are mentioned below:
Property 1: Perpendicularity
A tangent to a circle is perpendicular to the radius at the point of tangency.
Property 2: Equal Tangents
Tangents drawn from a common external point to a circle are equal in length.
Property 3: Angle Between Tangents
The angle between two tangents drawn from a common external point is bisected by the line segment joining the center of the circle to the external point.
Property 4: Tangent-Secant
The square of the length of a tangent drawn from an external point to a circle is equal to the product of the lengths of the secant segment and its external segment from the same point.
Tips and Tricks for Properties of Tangents to a Circle
Students often find the properties of tangents to a circle confusing. To avoid such confusion, we can follow these tips and tricks:
Perpendicularity to the Radius: Students should remember that a tangent to a circle is always perpendicular to the radius at the point of tangency. They can verify this by drawing a line from the circle's center to the tangent point and observing the right angle formed.
Equal Tangents: Students should recall that tangents drawn from a common external point are equal in length. To verify this, students can measure the tangents from the same external point to different points of tangency on the circle.
Tangent-Secant Relationship: Students should remember that the square of the tangent length equals the product of the whole secant segment and its external segment from the same external point.
Confusing Tangents with Secants
Students should remember that a tangent touches the circle at one point, whereas a secant intersects the circle at two points.
Mistake 1
Misinterpreting Perpendicularity
Students should know that a tangent is perpendicular to the radius at the point of tangency and not at any other point on the circle.
Mistake 2
Incorrectly Applying the Tangent-Secant Relationship
Students should practice the tangent-secant relationship: the square of the tangent length equals the product of the whole secant segment and its external segment.
Mistake 3
Misunderstanding Equal Tangents
Students should remember that tangents from a common external point are equal, not from different external points.
Mistake 4
Forgetting the Point of Tangency Rule
Students must remember that the point where the tangent touches the circle is called the point of tangency, and it is crucial for determining perpendicularity to the radius.
Mistake 5
Solved Examples on the Properties of Tangents to a Circle
A circle has a center O, and a tangent at point A is drawn from point P. If OP = 10 cm and OA = 6 cm, find the length of PA.
PA = 8 cm.
Problem 1
Using the Pythagorean theorem, since OP is the hypotenuse and OA is perpendicular, PA = √(OP² - OA²) = √(10² - 6²) = √(100 - 36) = √64 = 8 cm.
Two tangents PA and PB are drawn from an external point P to a circle with center O. If PA = 15 cm, what is the length of PB?
Explanation
PB = 15 cm.
Problem 2
Since tangents drawn from a common external point are equal, PA = PB. Thus, PB = 15 cm.
In a circle, a tangent at point A meets a line segment OP where O is the center of the circle. If OA = 5 cm and OP = 13 cm, find the length of PA.
Explanation
PA = 12 cm.
Problem 3
Using the Pythagorean theorem, since OP is the hypotenuse and OA is perpendicular, PA = √(OP² - OA²) = √(13² - 5²) = √(169 - 25) = √144 = 12 cm.
If a tangent from a point P to a circle is 9 cm long and the distance from P to the center of the circle is 15 cm, what is the radius of the circle?
Explanation
Radius = 12 cm.
Problem 4
Using the Pythagorean theorem, since OP is the hypotenuse and PA is perpendicular, Radius = √(OP² - PA²) = √(15² - 9²) = √(225 - 81) = √144 = 12 cm.
A circle has center O, and two tangents PA and PB are drawn from point P such that PA = 6 cm. If angle APB = 60 degrees, what is angle AOB?
Explanation
Angle AOB = 120 degrees.
A tangent to a circle is a straight line that touches the circle at exactly one point.
1.How many tangents can be drawn from a point outside a circle?
2.Are tangents from the same external point equal?
3.How do you find the length of a tangent?
4.Can a tangent pass through the circle?
Common Mistakes and How to Avoid Them in Properties of Tangents to a Circle
Students tend to get confused when understanding the properties of tangents to a circle, and they often make mistakes while solving problems related to these properties. Here are some common mistakes and solutions 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.
