CBSE Class 10 Maths Notes Chapter 10 Circles are an excellent resource, for knowing a particular chapter’s concepts in a crisp, friendly manner. Our articles help to learn children in their language, with proper images and solved examples for a better understanding of the concepts.
Chapter 10 of the NCERT Class 10 Maths textbook delves into the world of Circles, and their tangent theorems. It covers various topics such as introduction to tangents, properties of tangents, and theorems for circles on tangents These notes are designed to give students a comprehensive summary of the entire chapter and include all the essential topics, formulae, and concepts needed to succeed in their exams.
Need of Studying Circles
A circle can be found anywhere in real-life. Circles properties, can be applied in physics, for example, a tyre rotating on its axis, Earth is a circle if viewed in 2d. If multiple circles are combined, they form a sphere. This chapter covers, how tangent lines have some unique properties with circles.
Circle and Its Parts
Before starting our discussion, we will discuss the commonly used terminologies in circles. These terminologies are important, because the examiner uses these terms, to define the problem statement.
Circle and Center: Circle is a collection of points, such that the distance between the collection of points is equal to a given point, name centre. It is a 2-D shape that has no edges, and vertices.
Radius: The radius is the distance between the center of the circle, and any point on the circle. The radius of a circle is half of the diameter of the circle.
Chord: A chord is a line segment, whose endpoints, always lie on the circle. The largest chord in the circle is the diameter of the circle.
Secant: A secant is a line, passing through the circle, and intersects at 2 points, in the circle.
Note: Chord and Secant are two different things. Chord is a line segment, and secant is a line, with no end points.
Sector: A sector is an area enclosed, in an arc, and two radii of the circle.
Note: Area of a sector can be found by using the below formula
Area of full circle, i.e., for 360°angle = πr2
Area of a unit degree circle i.e., for 1°angle = πr2/​360°
Tangent: A tangent is a line that intersects at one point in the circle. Tangent is also a secant, where the points of intersection are the same.
Position of Line in a circle
There could be three possibilities, according to which the line is referenced, and the number of intersection points is defined.
Line Outside the Circle: A line outside the circle, do not touch the line at any position on the circle. It is said to be a non-intersecting line and has 0 points of intersection.
Line at the Circle: A line at the circle is called a tangent and has one intersection point.
Line Inside the Circle: A line passing through the circle cuts the circle at two positions and is said to be a secant line.
Position of Point in a Circle
There could be three possibilities, according to which the number of tangents passing is defined.
Point Inside the Circle: If we look at the image below, then, we will observe that the number of tangents, passing, is 0, and an infinite number of secants can be passed through that point.
Point at the Circle: As the point is on the circle, only one tangent can pass through the circle.
Point Inside the Circle: If a point lies outside the circle, then, exactly two tangents can pass through the circle. This property will be very significant in solving problems, which is mentioned in the later stage of the article.
Circle Theorems for the Tangents
The above discussion was important to understand the theorems related to circles. Only two theorems have been added to the syllabus, which defines the properties of tangents.
Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Note: The point of contact is the intersection point, where radius, and tangent of a circle meet.
Explanation:
A circle has infinitely many lines of symmetry. Let us consider a line, passing the centre of the circle. It cuts the circle into halves, and at the point of contact, a tangent is formed. Due to the symmetry, there is no reason to assume that a > b or b > a. Hence, radius and tangent form right angles.
Theorem 2: The lengths of tangents drawn from an external point to a circle are equal.
Explanation:
An external Point A cuts the circle at two points, B, and C. Line AB, and AC, are tangents to the circle. From theorem 1, as radius, and tangent intersect at right angles, make line OB, and OC, where ∠OBA = ∠OCA = 90°.
By Concurrency of Triangles:
In â–³ABO and â–³ACO,
⇒ AO = AO (Common)
⇒ OB = OC (Radius)
⇒ ∠ABO = ∠ACO (right-angle each)
By RHS, △ABO ≅ △ACO (Congruence Triangle)
Hence, AB = AC (Length of tangent is equal)
Note: ∠BOA = ∠AOC, as above triangles are congruent.
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Practice Problems on Circles
All the concepts required for this chapter have been covered, but that won’t be enough for this chapter, practicing questions is necessary, some questions, have been solved below for your practice.
Problem 1: From point Q, the length of the tangent to a circle is 12 cm and the distance of Q from the centre is 25 cm. The radius of the circle is?
Solution:
Step 1: Draw the diagram, as per the question.
Step 2: Write the given information, OB = 25cm, AB = 12cm, OA = ?
Step 3: Apply, the theorems, and find the required answer. Below is the calculation shown.
In△OAB, As per theorem 1,∠OAB=90°
Apply Pythagoras theorem in â–³OAB,
=>OA2 + AB2 = OB2
=>OA2 = 252 − 122
=>OA = √481
​=>OA = 21.93cm
The radius of the circle is 21.93cm.
Problem 2: As per the figure shown, if AC and BC are the two tangents to a circle with centre O so that ∠AOB = 100°, then ∠ACB is equal to?
Solution:
Step 1: The figure has already been provided. As, we need to find ∠ACB, join line OC.
Step 2: Write the given information, ∠AOB = 90°.
Step 3: Apply, the theorems, and find the required answer. Below is the calculation shown.
From theorem 1, ∠OAC = ∠OBC = 90°
As, △AOC≅△BOC
∠AOC=∠BOC=50°,
∠ACO=∠BCO=x,
∠OAC+∠AOC+x=180°
x=180−90−50
x=40°
∠ACB=2x=80°
FAQs on NCERT Notes for Class 10 Maths Chapter 10 Circles
Q1: What is a circle?
Answer:
A circle is the collection of points, such that the distance between the set of points, and a given point is constant. A circle can also be defined as a 2-D shape, which has no vertices, and edges.
Q2: When a secant is said to be tangent ?
Answer:
A secant, is the line passing passing inside the circle. A secant, cuts circle at two points. A secant is said to be tangent, if the two points of intersections of the secant are same i.e. secant, cuts the circle at only one point.
Q3: What is the maximum number of parallel tangents a circle can have?
Answer:
The maximum number of parallel tangents a circle can have is 2. Parallel tangents are formed, if perpendicular lines are drawn, at the end points of a diameter.
Q4: What is the General Equation of a circle with center (0, 0), and radius r?
Answer:
The general equation of a circle with center (0, 0), and radius r is,
x2 + y2 = r2
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