NCERT Solutions for Class 10 Mathematics Chapter 10 Circles

NCERT Solutions for Class 10 Mathematics Chapter 10 - Circles

Welcome to the Chapter 10 - Circles, Class 10 Mathematics NCERT Solutions page. Here, we provide detailed question answers for Chapter 10 - Circles. The page is designed to help students gain a thorough understanding of the concepts related to natural resources, their classification, and sustainable development.

Our solutions explain each answer in a simple and comprehensive way, making it easier for students to grasp key topics Circles and excel in their exams. By going through these Circles question answers, you can strengthen your foundation and improve your performance in Class 10 Mathematics. Whether you’re revising or preparing for tests, this chapter-wise guide will serve as an invaluable resource.

Exercise 1

Q 1:

How many tangents can a circle have?

Circle is the locus of points equidistant from a given point, which is the centre of the circle. And, tangent is the line which intersects a circle at one point only. On these points which touches at only one point. Hence, a circle can have infinite tangents.

Q 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 .

A tangent to a circle intersect it in one point.
A line intersecting a circle in two points is called a secant.
A circle can have two parallel tangent s at the most.
The common point of a tangent of a circle and the circle is called point of contact.

Q 3:

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. Length PQ is:

Q 4:

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.

In the given figure , AB and BC are two parallel lines. The line segment BC is the tangent at point x while AB is the secant to the circle.

Exercise 2

Q 1:

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. The radius of the circle is
(A) 7 cm (B) 12 cm
(C) 15 cm (D) 24.5 cm

Diagram drawn according to the question. Here OP is the radius of the circle.

Q 10:

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

Q 11:

Prove that the parallelogram circumscribing a circle is a rhombus.

Q 2:

In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠ POQ = 110°, then ∠ PTQ is equal to
(A) 60° (B) 70°
(C) 80° (D) 90°

Q 3:

If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠ POA is equal to
(A) 50° (B) 60°
(C) 70° (D) 80°

Q 4:

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

Q 5:

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

Q 6:

The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

Q 7:

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.

Q 8:

A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that AB + CD = AD + BC

Given : A quadrilateral ABCD which circumscribe a circle .

To prove: AB + CD = AD + BC

Proof: As we know tangents drawn from external point are equal. Therefore, we have

DR = DS …………….. (1)

AP = AS ……………… (2)

PB = BQ ……………… (3)

CR = CQ ………………. (4)

Adding equation (1), (2), (3) and (4), we get

DR + AP + PB + CQ = DS + AS + BQ + CR

AB + CD = AD + BC

Hence proved.

Q 9:

In Fig. 10.13, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that ∠ AOB = 90°.

Frequently Asked Questions about Circles - Class 10 Mathematics

- 1. How many questions are covered in Circles solutions?
- All questions from Circles are covered with detailed step-by-step solutions including exercise questions, additional questions, and examples.
- 2. Are the solutions for Circles helpful for exam preparation?
- Yes, the solutions provide comprehensive explanations that help students understand concepts clearly and prepare effectively for both board and competitive exams.
- 3. Can I find solutions to all exercises in Circles?
- Yes, we provide solutions to all exercises, examples, and additional questions from Circles with detailed explanations.
- 4. How do these solutions help in understanding Circles concepts?
- Our solutions break down complex problems into simple steps, provide clear explanations, and include relevant examples to help students grasp the concepts easily.
- 5. Are there any tips for studying Circles effectively?
- Yes, practice regularly, understand the concepts before memorizing, solve additional problems, and refer to our step-by-step solutions for better understanding.

Exam Preparation Tips for Circles

The Circles is an important chapter of 10 Mathematics. This chapter’s important topics like Circles are often featured in board exams. Practicing the question answers from this chapter will help you rank high in your board exams.

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