Class 10 — Mathematics (NCERT)
Circles: Tangents and Number of Tangents from a Point
Class 10
- ✓By the end of this lesson students will be able to define a tangent to a circle and identify its point of contact.
- ✓By the end of this lesson students will be able to state and apply the theorem that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
- ✓By the end of this lesson students will be able to determine the number of tangents that can be drawn to a circle from a point based on its position (inside, on, or outside the circle).
- ✓By the end of this lesson students will be able to state and apply the theorem that the lengths of tangents drawn from an external point to a circle are equal.
- ✓By the end of this lesson students will be able to solve problems involving tangents to a circle using geometric properties and the Pythagoras theorem.
Key concepts
A tangent to a circle is a line that intersects the circle at exactly one point. This unique point is called the point of contact. A tangent never enters the interior of the circle.
The tangent at any point of a circle is perpendicular to the radius through the point of contact. If a line AB is tangent to a circle with centre O at point P, then the radius OP is perpendicular to the tangent AB. This means ∠OPA = 90°.
The number of tangents that can be drawn from a point to a circle depends on the position of the point relative to the circle:\n1. If the point lies inside the circle, no tangent can be drawn through it.\n2. If the point lies on the circle, exactly one tangent can be drawn through it.\n3. If the point lies outside the circle, exactly two tangents can be drawn from it to the circle.
The lengths of tangents drawn from an external point to a circle are equal. If P is an external point and PQ and PR are two tangents drawn from P to a circle with centre O, then PQ = PR.
Key facts to remember
- 1A tangent is a line that intersects a circle at exactly one point, called the point of contact.
- 2Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
- 3No tangent can be drawn to a circle from a point lying inside it.
- 4Exactly one tangent can be drawn to a circle from a point lying on it.
- 5Exactly two tangents can be drawn to a circle from a point lying outside it.
- 6Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.
- 7The Pythagoras theorem is frequently used in problems involving tangents and radii, as they often form right-angled triangles.
Worked examples
Example 1
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Find the length of PQ.
Answer
The length of PQ is √119 cm.
Always draw a neat diagram to visualise the problem and identify the right-angled triangle.
Example 2
From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. Find the radius of the circle.
Answer
The radius of the circle is 7 cm.
Remember that the radius is always perpendicular to the tangent at the point of contact, forming a right-angled triangle.
Example 3
Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
Answer
The length of the chord of the larger circle is 8 cm.
This problem combines the tangent-radius perpendicularity with the property of a chord being bisected by a perpendicular from the centre.
Common mistakes
- ✗Confusing a secant (a line intersecting a circle at two points) with a tangent.
- ✗Incorrectly assuming that the angle between a tangent and a chord through the point of contact is 90 degrees, instead of the angle between the tangent and the radius.
- ✗Applying Pythagoras theorem incorrectly by misidentifying the hypotenuse in a right-angled triangle.
- ✗Forgetting that the perpendicular from the centre to a chord bisects the chord, which is often crucial in combined problems.
- ✗Not drawing a clear and labelled diagram, leading to errors in understanding the relationships between lines and points.
Exam tips
- ★Always draw a neat and labelled diagram for every problem. This helps in visualising the given information and identifying relevant geometric figures, especially right-angled triangles.
- ★Clearly state the theorems you are using in your solution (e.g., 'By Theorem 10.1, OP ⊥ PQ'). This shows your understanding and earns marks.
- ★Look for right-angled triangles formed by radii, tangents, and distances from the centre. The Pythagoras theorem is a very common tool in these problems.
- ★Practice solving a variety of problems to become proficient in applying both Theorem 10.1 and Theorem 10.2, as well as other circle properties learned previously.
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