CAT 2025 Lesson : Circles - Terminologies

1. Introduction

Most questions from geometry involve circles in some form. A lot of the questions are combinations with triangles and/or quadrilaterals. Please read through the Lines & Triangles and Quadrilaterals lessons before you commence this lesson.

In this lesson we will begin with defining the basic terminologies and then move on to properties of circles including areas, chords, secants, tangents, etc.

2. Basic Terminologies

Terminologies	Figure
Circle: A circle is a shape formed by a point that moves such that its distance from a certain point in the middle of the circle is constant. This point is called the Centre of the circle. O is the centre of this circle.
Arc: An arc is the part of the circle between 2 points on its circumference. The smaller length is called Minor Arc and the longer length is called Major Arc
Radius: The line drawn from the centre of the circle to any point on the circle is called the radius. The plural of radius is radii. OA is the radius of the circle with centre O.
Sector: A sector is the area enclosed by $2$ radii and the arc of the circle. The smaller area is called the minor sector and the larger area is called the major sector. The shaded region is the minor sector and the unshaded region is the major sector.
Secant: A secant is a line that intersects the circle at exactly $2$ points. It divides the circle into two Segments. The segment with smaller area is called the minor segment and the larger segment is the major segment $l$ is a secant. The shaded region is the minor segment and the unshaded region is the major segment.
Chord: A chord is a line segment that has both its end points on the circumference of a circle. It is the part of the secant on and inside the circle. AB is a chord of this circle.
Diameter: The diameter is the longest chord of a circle that passes through the centre. It is twice the radius. It divides the circle into $2$ equal segments called semi circles. AC is a diameter of this circle.
Tangent: A tangent is the line that meets or touches the circle at exactly one point. $t$ is a tangent that touches this circle at point A.

3. Additional Terminologies

Terminologies	Figure
Inscribed Angle: The angle subtended by two chords which meet on the circumference of a circle is the inscribed angle. $\angle$ ADB is the angle inscribed by the chords AD and BD.
Central Angle: The angle subtended by $2$ radii at the centre of the circle is the central angle. Measure of an arc: Angle formed by an arc at the centre. Measure of Minor arc will be $< 180^\mathrm{o}$ , while the measure of a Major Arc will be $> 180^\mathrm{o}$ . If the measure of minor arc is $x^\mathrm{o}$ , then that of the major arc will be $360^\mathrm{o} - x^\mathrm{o}$ . $\angle$ AOB is the central angle and measure of the arc AB.
Congruent Circles: Circles with equal radii are called Congruent circles. Note that all circles are similar to each other. In this figure, OB $=$ PK. Therefore, the $2$ circles are congruent.
Concentric Circles: Circles with a common centre are called concentric circles. As all these circles have a common centre, i.e., O, they are concentric circles.
Concyclic Points: Points around which a circle can be drawn are called concyclic points. Here, A, B and C are concyclic, whereas O and D are not. *(Add a point D outside the circle)*
Circumcircle: A circle that can pass through all the vertices of a polygon is called a circumcircle of that polygon. A circumcircle can be drawn for for all triangles and regular polygons. However, for polygons with $4$ or more sides, a circumcircle can be drawn only if they are cyclic polygons (i.e., where the vertices of the polygon are concyclic). PQRSTU is a regular hexagon around which a circle has been circumscribed. If a quadrilateral can be circumscribed by a circle, then it is called a cyclic quadrilateral. ABCD is a cyclic quadrilateral. ACD is a triangle around which a circumcircle has been drawn.
Incircle: A circle that is inscribed in a polygon and touches all the edges of the polygon at exactly one point is an incircle. Like a circumcircle, an incircle is possible for all triangles and regular polygons. However, for polygons with $4$ or more sides, an incircle can be drawn only if they are tangential polygons (i.e. Where the sides of the polygons are tangents to the circle). LMN is the triangle which has a circle inscribed in it. KLMN is a square which has a circle inscribed in it. Therefore, KLMN is a tangential quadrilateral. Note that all square are tangential quadrilaterals.

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