NCERT Solutions Class 10 Maths Chapter 10 Circles

Circles is the tenth chapter of the NCERT Class 10 Maths curriculum, offering comprehensive coverage of key geometric concepts. This chapter includes important topics such as the introduction to circles, the tangent to a circle, the number of tangents from a point on a circle, and a summary. The chapter starts with an introduction in section 10.1 and progresses to more detailed discussions in sections 10.2, 10.3 and 10.4. For thorough practice, students can refer to NCERT Solutions of Class 10 Circles Exercise 10.1 and Class 10 Circles Exercise 10.2, which provides practical problems based on the concepts covered. Practising these exercises will help students build a solid understanding of this chapter, which is essential for excelling in their exams.

1.0Download NCERT Solutions for Class 10 Maths Chapter 10 Circles : Free PDF

ALLEN's experts have meticulously crafted the Class 10 Circles solutions to enhance students' problem-solving skills. To gain a clear understanding of the NCERT Solutions for Circles, students can download the PDF from the link below:

2.0NCERT Solutions Class 10 Maths Chapter 10 Circles: Important Topics

• Introduction to Circles: Introduces the basic properties and definitions of circles, laying the foundation for further exploration of circular geometry.
• The Tangent to a Circle: Explains the concept of a tangent, its properties, and how it relates to a circle at a single point.
• The Number of Tangents from a Point on a Circle: Discuss how to calculate the number of tangents that can be drawn from an external point to a circle.
• Summary: Recaps all essential concepts and formulas from the chapter, providing a comprehensive review for better retention.

3.0NCERT Class 10 Maths Chapter 10 Circles: Exercise Solutions

There are two exercises in chapter 10 (Circles) of class 10 Maths. Students can find the split below:

Explore Circles exercises solutions and learn how to solve various problems only on NCERT Solutions For Class 10 Maths.

4.0Advantages of Studying Class 10 Maths Chapter 10 NCERT Solutions –  Circles

1. Strengthens Geometry Skills: Enhances understanding of geometric concepts and their practical applications.
2. Foundation for Higher Studies: Builds a strong base for advanced mathematical concepts in higher classes.
3. Improves Problem-Solving Ability: Encourages logical thinking and helps in tackling real-world geometry problems.
4. Scoring Chapter: If concepts are clear, it's a relatively easy chapter to score well in exams.

5.0Important Topics of Class 10 Math Chapter 10 Circles Solutions

Chapter 10, Circles, focuses on the properties and theorems related to circles, including tangents, chords, and secants. Key concepts include the tangent being perpendicular to the radius, the number of tangents from an external point, and the Intersecting Chords Theorem. Students also learn important angle properties, such as the angle subtended by a chord at the center and the angle subtended by a diameter. This chapter helps in understanding geometric properties and their applications in solving problems.

1. Tangent to a Circle
2. Number of Tangents from an External Point
3. Theorem on Tangent-Secant and Tangent-Tangent
4. Intersecting Chords Theorem
5. Theorems Related to Angles in a Circle

6.0General Outline for Class 10 Maths Chapter 10: Circles

Understanding Key Concepts:

• Circle: A circle is the set of all points in a plane that are at a fixed distance (radius) from a fixed point (center).
• Radius: The distance from the center of the circle to any point on the circle.
• Diameter: A line segment passing through the center of the circle, with both ends on the circle. The diameter is twice the radius (D = 2r).
• Chord: A line segment joining two points on the circle.
• Secant: A line that intersects the circle at two points.
• Tangent: A line that touches the circle at exactly one point (point of contact).

Important Theorems and Properties:

• Theorem 1: The tangent to a circle is perpendicular to the radius at the point of contact.
• Theorem 2: If two tangents are drawn to a circle from an external point, then the tangents are equal in length.
• Theorem 3: The angle subtended by a chord at the center of the circle is twice the angle subtended on the remaining part of the circle (Angle at the center theorem).
• Theorem 4: The angle subtended by a chord at any point on the circle is constant (Angle subtended by the same chord theorem).
• Theorem 5: In a circle, the perpendicular from the center to a chord bisects the chord.

Key Formulas:

• Length of a Tangent from an External Point: If the distance from the external point to the center of the circle is d and the radius of the circle is r, the length of the tangent tt is given by: $t=\sqrt{d^2-r^2}$
• Angle Between Two Tangents: The angle between two tangents drawn from an external point is given by: $\theta =2\times \text{angle subtended by the chord at the center}$
• Perpendicular from the Center to a Chord: The perpendicular from the center bisects the chord, so if the length of the chord is 2a and the perpendicular distance from the center is d, then:$r^2=a^2+d^2$ where r is the radius of the circle.

7.0Sample NCERT Solutions for Class 10 Maths Chapter 10 Circles

1. How many tangents can a circle have?

Sol. There can be infinitely many tangents to a circle.

2. Fill in the blanks :

(i) A tangent to a circle intersects it in ..... point (s).

(ii) A line intersecting a circle in two points is called a.......

(iii) A circle can have ……… parallel tangents at the most.

(iv) The common point of a tangent to a circle and the circle is called …….

Solution:

(i) One

(ii) Secant

(iii) Two

(iv) Point of contact.

3. Draw a circle and two lines parallel to a given line such that one is a tangent and the other a secant to the circle.

Solution:

We have the required figure as shown:

Here, l is the given line and a circle with center O is drawn.

The line n is drawn which is parallel to l and tangent to the circle. Also, m is drawn parallel to line l and is a secant to the circle.

4. 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. The radius of the circle is -

(A) 7 cm

(B) 12 cm

(C) 15 cm

(D) 24.5 cm

Solution:

From figure,

r² = (25)² - (24)²

= 625 - 576 = 49

r = 7 cm

Hence, the correct option is (A).

5. If tangents PA and PB from a point P to a circle with centre O are inclined to each other at an angle of 80°, then ∠POA is equal to:

(A) 50°

(B) 60°

(C) 70°

(D) 80°

Solution:

In the figure,

ΔOAP ≅ ΔOBP (SSS congruence)

⇒ ∠POA = ∠POB = (1/2)∠AOB  ...(i)

Also, ∠AOB + ∠APB = 180°

⇒ ∠AOB + 80° = 180°

⇒ ∠AOB = 100° ...(ii)

Then, from (i) and (ii),

∠POA = (1/2) × 100° = 50°

Hence, the correct option is (A).

8.0Benefits of Studying Class 10 Maths Chapter 10 Circles

The Circles chapter in Class 10 Maths offers several key benefits:

1. Foundation for Advanced Geometry: It introduces important concepts like tangents, chords, and cyclic quadrilaterals, which are crucial for higher geometry studies.
2. Real-Life Applications: Concepts from this chapter are used in fields like engineering, architecture, and design, especially in circular structures and objects.
3. Problem-Solving Skills: Students enhance their ability to solve geometric problems involving circles and understand their properties.
4. Visualizing Shapes: It helps students visualize and understand the relationship between various parts of a circle, improving spatial awareness.

In short, the Circles chapter strengthens geometric reasoning, problem-solving skills, and has practical applications in real-world scenarios.