Circles

# Class 10 Maths Chapter 15 Circles Important Questions

Chapter 15 of ICSE Class 10 Mathematics, titled "Circles," is a fundamental chapter that delves into the geometry of circles. Circles are one of the most important geometric shapes and have numerous applications in mathematics and real-life situations. This chapter introduces students to the properties, theorems, and concepts related to circles, enabling them to solve problems involving circles with confidence. Questions on circles for class 10 ICSE are designed as a valuable tool to help students build confidence in their mathematical skills, alleviate doubts, and overcome difficulties they might encounter while studying this topic. students can strengthen their grasp of this fundamental mathematical concept, making it easier for them to excel in their class 10 Mathematics exams.

## Introduction

Circles of class 10 ICSE are one of the fundamental geometric shapes that have been studied and admired for their simplicity and symmetry since ancient times. In ICSE Class 10 Mathematics, the chapter on circles introduces students to the various aspects of circles, including their definitions, properties, and theorems. Understanding circles is not only crucial for solving geometric problems but also for comprehending more advanced mathematical concepts.

Key Concepts Covered in the Circles Chapter:

Definitions: The chapter begins with the basic definitions of a circle, its center, radius, and diameter. Students learn to distinguish between various parts of a circle, such as arcs, chords, and secants.
Properties of Circles: Students delve into the fundamental properties of circles, including the fact that all points on the circumference are equidistant from the center. They also learn about central angles, inscribed angles, and the relationship between arc lengths and angles.
Circumference and Area: The chapter covers formulas for calculating the circumference and area of a circle. Students practice using these formulas to find measurements in real-world scenarios.

## What are Circles?

Circles of class 10 ICSE, Think of a circle as a perfectly round shape, like a pizza or a bicycle wheel. It's made up of points on the edge, and all these points are the same distance away from a special point right in the middle, which we call the center.

The center is like the heart of the circle, and the distance from this center to any point on the edge is called the radius. So, if you were to stretch your arm from the center to the edge, that's your radius!

Diameter - Twice the Fun:
Now, if you take a line that goes through the center and touches the circle's edge on both sides, we call that line the diameter. It's like the circle's widest smile. Fun fact: The diameter is always twice as long as the radius. That's a cool math rule (D = 2r)!

Circumference - Circle's Border:
The circumference is like the circle's border or the path you would travel if you walked all the way around it. You can find the circumference by multiplying the diameter by a special number called π (pi). So, it's C = 2πr or C = πd.

Chords - Connecting Dots:
If you join any two points on the circle's edge with a straight line, that's called a chord. It's like connecting dots on a circle.

Arcs and Sectors - Pizza Slices:
Imagine cutting a pizza into slices. Each slice has a curved edge, which we call an arc. The area inside a slice, including the curved part and the lines from the center to the edge, is called a sector.

### Class 10 Circles Important Questions and Answers

#### (a) 45°(b) 65°(c) 75°(d) 55°

Ans. (b) 65°

Explanation:
since, Angle subtended by an arc at the centre is twice the angle subtended in the remaining part of the circle.

∵ ∠POQ =2∠QRP
⇒∠QRP =$$\frac{1}{2}$$  ∠POQ
=$$\frac{1}{2}$$ × 130°×60°
∵ OP = OR                                (same radius)
⇒∠OPR ×∠ORP= 60°

#### (a) Point of contact(b) Secant(c) Both (a) and (b)(d) None of these

Ans. (a) Point of contact

#### Q3. If ABCD is a cyclic quadrilateral in which AD || BC, prove that ∠B=∠C.

Explanation:
ABCD is a cyclic quadrilateral.
So,              ∠ A + ∠ C = 180° …(i)
Since             AD || BC
So,               ∠ B + ∠ A = 180° …(ii)
From equations (i) and (ii),
∠ A + ∠ C = ∠ B + ∠ A
⇒         ∠ C = ∠ B
or         ∠ B = ∠ C.

#### Q4. If AB and CD are two chords which when produced meet at P and if AP = CP, show that AB = CD.

Explanation:
Here, chords AB and CD of the circle intersect at P.
∴ PA × PB = PC × PD
PB = $$\frac{PC×PD}{PA}$$

⇒  PB = $$\frac{AP×PD}{AP}$$
{˙.˙PC = AP(given)}
⇒          PB = PD…(i)
Now,          AB = AP–BP
⇒          AB = CP–PD
[˙.˙AP = CP(given),BP=PDFrom(i)]
But          CD = PC–CP
Hence,          AB = CD.

#### Q5. In the figure, AC is the diameter of the circle, centre O. Chord BD is perpendicular to AC. Write down the angles p,q,r in term of x.

Explanation:
In the given circle,
⇒ ∠ ADB = $$\frac{1}{2}$$ ∠ AOB =$$\frac{x}{2}$$
⇒      ∠ ADB=∠ ACB=q
Combining these, we get
$$\frac{x}{2}$$= 90° − r = q
⇒      2r=180°–x
and      x=2q
∠ DAC=∠ CAB
=∠ BDC
⇒ p = r =$$\frac{1}{2}$$(180° x)

#### ICSE Class 10 Maths Chapter wise Important Questions

Chapter No. Chapter Name
Chapter 1 Goods and Service Tax (GST)
Chapter 2 Banking
Chapter 3 Shares and Dividends
Chapter 4 Linear inequations
Chapter 5 Quadratic Equations in one variable
Chapter 6 Ratio and proportion
Chapter 7 Factorization
Chapter 8 Matrices
Chapter 9 Arithmetic Progression
Chapter 10 Geometric Progression
Chapter 11 Coordinate Geometry
Chapter 12 Reflection
Chapter 13 Similarity
Chapter 14 Loci
Chapter 15 Circles
Chapter 16 Constructions
Chapter 17 Mensuration
Chapter 18 Trigonometry
Chapter 19 Statistics
Chapter 20 Probability

#### Conclusion

In Circles of class 10 ICSE, the study of circles in Class 10 ICSE mathematics is a fundamental part of geometry. Circles are fascinating shapes with unique properties and theorems that help us understand and solve various geometric problems. If you seek additional practice and a deeper comprehension of the topics covered in the chapter, oswal.io offers an extensive array of questions on circles for class 10 ICSE to facilitate a more profound understanding of the concepts.

#### Q1 : What is a circle?

Ans: A circle is a closed curve where all points on the edge are equidistant from a fixed point called the center.

#### Q2: What is the radius of a circle?

Ans:  The radius is the distance from the center of the circle to any point on its edge.

#### Q3 : How is the diameter related to the radius?

Ans: The diameter is always twice the length of the radius (D = 2r).

#### Q4 : What is the formula for finding the circumference of a circle?

Ans: The formula is C = 2πr or C = πd, where 'C' is the circumference, 'r' is the radius, and 'd' is the diameter.

#### Q5 : What is an arc in a circle?

Ans: An arc is a part of the circumference of a circle.

## Chapter Wise  Important Questions for ICSE Board Class 10 Mathematics

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