Class 10 Ex 10.2 Solutions

Class 10 Ex 10.2 Solutions: A complete walkthrough to Circles

This article provides comprehensive solutions for Exercise 10.2 of Class 10 mathematics, focusing on the properties of circles and their tangents. We'll cover each problem step-by-step, explaining the concepts involved and providing clear, concise solutions. Understanding these concepts is crucial for mastering geometry and succeeding in higher-level mathematics. This guide will help you not only solve the problems but also develop a deeper understanding of the underlying principles.

Introduction to Circles and Tangents

Before diving into the solutions, let's refresh our understanding of key terms:

• Circle: A circle is a set of all points in a plane that are equidistant from a given point called the center. The distance from the center to any point on the circle is called the radius.

• Tangent: A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of tangency or point of contact. The tangent is always perpendicular to the radius at the point of contact.

• Secant: A secant is a line that intersects a circle at two distinct points.

• Theorem: A crucial theorem states that the lengths of tangents drawn from an external point to a circle are equal. This theorem forms the basis for many problems in Exercise 10.2.

Solving Class 10 Ex 10.2 Problems: Step-by-Step Solutions

Exercise 10.So 2 typically involves problems applying the properties of tangents and the theorem mentioned above. We'll solve several example problems, demonstrating different approaches and problem-solving strategies. In practice, remember to always draw a clear diagram to visualize the problem. This significantly aids in understanding and solving geometrical problems.

Problem Example 1: (Illustrative Problem - Specific problem numbers will vary based on textbook)

Problem Statement: Two tangents are drawn from an external point P to a circle with center O. If the tangents touch the circle at points A and B, and the angle AOB is 120 degrees, find the length of PA if OA = 3cm.

Solution:

1. Draw a Diagram: Draw a circle with center O. Draw tangents PA and PB from an external point P. Mark the points of contact A and B. Join O to A, O to B, and O to P.

2. Identify Relevant Properties: We know that OA and OB are radii, and PA and PB are tangents. So, angles OAP and OBP are 90 degrees (radius is perpendicular to tangent at the point of contact).

3. Apply Properties to Solve: In quadrilateral OAPB, the sum of angles is 360 degrees. We have:

   ∠OAP + ∠APB + ∠PBO + ∠BOA = 360° 90° + ∠APB + 90° + 120° = 360° ∠APB = 360° - 300° = 60°

4. Use Trigonometric Ratios (if necessary): In right-angled triangle OAP, we have:

   sin(∠OPA) = OA / OP sin(30°) = 3cm / OP OP = 6cm

5. Find the Length PA: In right-angled triangle OAP, we can use Pythagorean theorem or trigonometric ratios:

   Using Pythagoras: OP² = OA² + AP² 6² = 3² + AP² AP² = 36 - 9 = 27 AP = √27 = 3√3 cm

Which means, the length of PA is 3√3 cm.

Problem Example 2: (Illustrative Problem - Specific problem numbers will vary based on textbook)

Problem Statement: Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Solution:

1. Draw a Diagram: Draw a circle with center O and diameter AB. Draw tangents at points A and B.

2. Apply Properties: Let the tangents at A and B be denoted as l and m respectively. The radius OA is perpendicular to tangent l at A, and the radius OB is perpendicular to tangent m at B.

3. Prove Parallelism: Since OA and OB are radii and are collinear (forming the diameter), the angles formed between the diameter and tangents (∠OAl and ∠OBm) are both 90 degrees. This makes tangents l and m parallel because they are both perpendicular to the same line (diameter AB).

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Problem Example 3: (Illustrative Problem - Specific problem numbers will vary based on textbook)

Problem Statement: From a point Q, the length of the tangent to a circle is 24 cm, and the distance of Q from the center is 25 cm. Find the radius of the circle.

Solution:

1. Diagram: Draw a circle with center O. Let Q be an external point. Draw a tangent from Q touching the circle at point R. Join O to Q and O to R.

2. Apply Properties: OQ = 25 cm (distance from center to external point), QR = 24 cm (tangent length). OR is the radius (let's denote it as 'r').

3. Pythagorean Theorem: In right-angled triangle OQR, we have:

   OQ² = OR² + QR² 25² = r² + 24² 625 = r² + 576 r² = 625 - 576 = 49 r = 7 cm

So, the radius of the circle is 7 cm.

Further Exploration and Advanced Concepts

While Exercise 10.2 primarily focuses on the basic properties of tangents, understanding more advanced concepts can deepen your understanding. These include:

• Length of Tangents from an External Point: The theorem stating that the lengths of tangents drawn from an external point to a circle are equal is fundamental. Understanding its proof and application is crucial.

• Properties of Tangents and Chords: Explore the relationships between tangents and chords within a circle. This involves understanding the angles subtended by arcs and the properties of cyclic quadrilaterals.

• Construction of Tangents: Learn how to construct tangents to a circle from both internal and external points using geometric tools like a compass and straightedge.

• Application of Similar Triangles: In many problems, similar triangles can be used to solve for unknown lengths or angles. Recognizing similar triangles is a powerful problem-solving technique.

Frequently Asked Questions (FAQ)

Q: What is the difference between a secant and a tangent?

A: A secant intersects a circle at two points, while a tangent touches the circle at only one point.

Q: Why is the angle between a radius and a tangent always 90 degrees?

A: This is a fundamental property of circles and tangents. The shortest distance from a point (the center) to a line (the tangent) is always along a perpendicular line. The radius connecting the center to the point of tangency is the shortest distance, hence the 90-degree angle.

Q: Are tangents always perpendicular to the radius at the point of contact?

A: Yes, this is a defining property of a tangent.

Q: Can two tangents be drawn from a single point outside the circle?

A: Yes, two tangents can be drawn from a single point outside the circle, and their lengths will be equal.

Q: How can I improve my problem-solving skills in geometry?

A: Practice is key! Work through numerous problems, draw clear diagrams, and try to understand the underlying principles rather than just memorizing formulas. Seek help when needed, and don't be afraid to ask questions.

Conclusion

Mastering Class 10 Ex 10.This complete walkthrough provides a strong starting point for achieving success in this crucial area of mathematics. By working through the problems step-by-step and understanding the underlying concepts, you'll build a strong foundation in geometry. This leads to remember to practice regularly, and don't hesitate to review the fundamental definitions and theorems. And 2 requires a solid understanding of circles, tangents, and related theorems. That's why continue to explore advanced concepts and apply your knowledge to solve more complex problems. Good luck!