Class 10 Maths Chapter 10 Circles Notes & Solutions

We have studied quite a bit about circles in the IXth class, therefore, we are familiar with this concept. In class 10 maths chapter 10 circles, we will mostly learn about the tangents. Almost all questions that we will see in this chapter will be based on the concept of the tangent.

The circle is a very common figure around us, we can see it everywhere. From simple and tiny things like a 5 rupees coin to something as big as planate (though it’s actually a sphere), almost everything is in the shape of a circle. This shape is very important to us, this is why CBSE has made a separate chapter for circles. We will go through all the details that will be needed to solve even the most trickier questions in this chapter.

Class 10 Maths Chapter 10 Circles

If you want to make great architecture, then you must have a solid foundation, the same is true for learning. If you want to learn something then you must first learn all the basics about it. This is why we have given all the terminologies that you will need to learn in class 10 maths chapter 10 circles.

Circle: A circle is the collection of all the points in a plane, which are at a fixed distance from a fixed point in the plane. The fixed point from which the distance is measured is called the centre, and the fixed distance is called radius (we will see the definition of radius later in this article).

Chord: It is a line segment that connects two points on the circumference of the circle.

Circumference of the circle: Circumference which is also known as the perimeter is simply the measurement of the boundary of the circle.

Diameter: The longest chord in a circle is called the diameter of a circle. The formula to find the diameter of a circle is 2×radius.

Radius: The fixed distance between the centre and a point in the circumference of a circle is called the radius of the circle. The radius of a circle is always exactly half of its diameter, the formula (with respect to diameter) is given by \frac{1}{2}\times diametre. The plural form of the radius is ‘radii’.

Arc: In class 10 maths chapter 10 circles, an arc is a piece of the circle between two points. The bigger piece of a circle is called the major arc and the smaller one is called the minor arc.

Semicircle: When a circle is divided by its diameter into two parts, then two arcs of equal size are obtained, both the arc will be called semicircles. In other words, a semicircle is exactly the half of a circle.

Segment: When a circle is divided by a chord in two unequal parts, then the region between the chord and
either of its arcs is called a segment. The area between the chord and bigger arc is called the major segment and, the area between the chord and smaller arc is called the minor segment.

Sector: The area between an arc and the two radii that connects the centre to the endpoints of the
arc is called a sector. The area between the two radii and the minor arc is called the minor sector, and the area between the radii and the major arc is called the major sector.

Class 10 Maths Chapter 10 Circles – Previous Class Summary

We now have an understanding of the basic terminologies that are used in chapter 10. But, we still need to brush up on our knowledge from the previous class. Let us go through the important points that we had studied in class 9th.

  • Equal chords of a circle or we can say equal chords of congruent circles subtend equal angles at the centre.
  • If a perpendicular is drawn from the centre of a circle to a chord then it will bisect the chord. Similarly, if a line is drawn through the centre of a circle to bisect a chord, then it will be perpendicular to that chord.
  • There can be only one circle passing through three non-collinear points. Suppose, there is a triangle, if we try to draw a circle that touches all three points of the circle, then only one circle can be drawn. This circle, its centre and radius will be called circumcircle, circumcentre, and circumradius of that triangle respectively.
  • Equal chords of a circle are at equal distance i.e equidistant from the centre.
  • If two arcs of a circle are congruent, then their corresponding chords are equal and
    conversely if two chords of a circle are equal, then their corresponding arcs (minor, major)
    are congruent.
  • For class 10 maths chapter 10 circles, we should know that congruent arcs of a circle subtend equal angles at the centre.
  • The angle that is subtended by an arc at the centre is always double the angle that it will subtend at any point on the remaining part of the circle.
  • Angles that are in the same segment of a circle are always equal.
  • Angle in a semicircle is a right angle.
  • If a line segment that is joining two points subtends equal angles at two other points that are on
    the same side of the line containing the line segment, then there will be four points on a circle.
  • In a cyclic Quadrilateral, the sum of either pair of opposite angles is always 1800. Therefore we can say that if the sum of a pair of opposite angles of a quadrilateral is 1800 then it is a cyclic quadrilateral.

Class 10 Maths Chapter 10 Circles – Explanation

Now that we have refreshed all the concepts and important points that we have studied about the circles so far, we can now proceed to actually learn new things about this chapter. There is not much theory to this chapter, but still, we will go through the important points that are worth noting in the following article.

If a line and a circle are given on a plane, then there can only be three conditions i.e
condition 1: the line does not touch the circle at all.
condition 2: line goes through the circle. In such a scenario, the line will be called a chord as it touches the circle at two points. The chord is also referred to as secant.
condition 3: The line will touch the circle at only one point (this point will be known as the point of contact). In this scenario, the line is called the tangent.

# Tangent to a circle 

A tangent to a circle is a line that intersects the circle at only one point. The word ‘tangent’ is derived from the Latin word ‘tangere’ meaning ‘to touch’. The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Thomas Fineke who was a Mathematician from Denmark first introduced the concept of tangent in 1583.

Note: We can draw infinite tangents on a circle, but,  there can only be one tangent at a point of the circle.

# Number of tangents from a point on a circle

An infinite number of tangents can be drawn to a circle, but, there can only be two parallel tangents on a circle. The lengths of tangents drawn from an external point to a circle are equal. Lets take a look at the blueprint of this chapter.

Class 10 maths chapter 10 circles blueprint
1 Marker 2 questions
2 Marker 1 question
Total questions 3
Total marks 4

As per the marking scheme released by CBSE this year, class 10 maths chapter 10 circles will be of total 4 marks for 3 questions. two 1 markers and one 2 markers will come in the board exam. The marking scheme may vary in the exam.

Updated: December 8, 2021 — 11:47 am

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