CBSE Class 10t Number of Tangents from a Point on a Circle Details & Preparations Downloads

In the dynamic landscape of Class 10 mathematics, the concept of the number of tangents from a point to a circle stands as a key pillar. This intriguing topic not only delves into the geometric intricacies of circles but also opens a gateway to understanding the symmetrical dance between points and tangents. Join us as we unravel the mysteries of this fundamental concept.

Unlocking Geometric Mastery: CBSE NCERT Download Deciphers the 'Number of Tangents from a Point on a Circle' in Class 10

I. The Fundamentals: Grasping the Basics

Before we embark on our journey, it's essential to establish a solid foundation. Students delve into the fundamental principles, understanding the nature of tangents and their unique relationship with circles. This sets the stage for a deeper exploration.

II. Single Tangent: The Simplicity of Unity

The blog explores scenarios where a single tangent extends from a point to a circle. Students discover the elegance in simplicity and the foundational principles that govern the connection between a point and its tangent on a circle.

III. Two Tangents: The Symmetry Unveiled

As we progress, the concept unfolds into scenarios involving two tangents from a point to a circle. Students explore the symmetrical nature of these tangents, unraveling the geometry that underlies the connection between points and the circular boundary.

IV. Three or More Tangents: Navigating Complexity

The exploration extends to scenarios where three or more tangents emerge from a single point to the circle. The blog dissects the geometric patterns that arise, empowering students to navigate the complexities of multiple tangents and their spatial relationships.

V. Problem-Solving Strategies: Applying Knowledge with Confidence

The journey concludes with a focus on problem-solving strategies. Armed with a deep understanding of the number of tangents from a point on a circle, students gain the tools to confidently approach and solve a variety of geometric challenges, enhancing their analytical and critical-thinking skills.

Download Mathematics notes

Tangents From a Point on a Circle Theorem and Proof

The two theorems related to the number of tangents from a point on a circle are explained in detail.
Theorem:
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Proof:
Consider a circle with centre “O” and XY is the tangent to a circle with the point of contact “P”.
To prove: OP is perpendicular to XY.
Take a point Q on the tangent XY other than point P. Now, join the points O and Q.
Note that point Q should lie outside the circle. Because, if the point Q lies inside the circle, XY will become a secant and there is no tangent to a circle.
So, we can say that OQ is greater than OP.
(i.e) OQ > OP
Since the point lies outside of the circle, the above condition is applicable for every point on XY, except the point P.
Thus, OP is the shortest distance of point O to the points on the tangent XY.
Therefore, OP is perpendicular to tangent XY.

Examples

Question:

Let TP and TQ be the two tangents drawn to a circle with centre O from an external point T. Show that ∠PTQ = 2 ∠OPQ.

Solution:

Assume a circle with centre O and the external point is T. Hence, the two tangents formed are TP and TQ to a circle where P and Q are the points of contact as shown in the figure.

To prove: ∠PTQ = 2 ∠OPQ.

Let ∠PTQ = θ.

Now, by using the theorem “the lengths of tangents drawn from an external point to a circle are equal”, we can say TP = TQ. So, TPQ is an isosceles triangle.

Therefore,

∠TPQ = ∠TQP = ½ (180°− θ ) = 90° – (½) θ

By using the theorem, “the tangent at any point of a circle is perpendicular to the radius through the point of contact”, we can say ∠OPT = 90°

Therefore,

∠OPQ = ∠OPT – ∠TPQ = 90° – [90° – (½) θ]

∠OPQ = (½)θ

∠OPQ = (½) ∠PTQ, which can also be written as:

∠PTQ = 2 ∠OPQ.

CBSE Class 10th Downloadable Resources:

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SAMPLE PRACTICE QUESTION

Q1: How many tangents can be drawn from a point outside a circle to the circle?

Ans: Two tangents can be drawn from a point outside a circle to the circle.

Q2: What is the significance of the two tangents drawn from an external point to a circle?

Ans: The two tangents from an external point to a circle are of equal length, and they form an isosceles triangle with the radius of the circle. The point of contact with the circle is the vertex of this triangle.

Q3: How does the length of the tangents relate to each other and to the radius of the circle?

Ans: The lengths of the two tangents are equal to each other and are also equal to the radius of the circle.

Q4: Can more than two tangents be drawn from a point outside a circle?

Ans: No, only two tangents can be drawn from a point outside a circle. This is a unique geometric property.

Q5: What is the relationship between the angle formed by the two tangents and the central angle subtended by the same arc?

Ans: The angle formed by the two tangents at the point of contact is equal to the central angle subtended by the same arc on the circle.