CAT Circles' basic concepts are one of the important topics of the geometry section in the quantitative ability section. It is part of geometry questions. Geometry questions contain 20% weightage in the CAT exam. However, Several direct and sometimes indirect questions have been asked in CAT Question Paper. But the basics of CAT Circles will help you in solving geometry questions which have 20% weightage.

In the past, many questions have been asked from CAT circles in the quant section. Many questions are not directly asked questions from circles. But they make use of circle concepts through some indirect concepts of circles.

Here we will get to know important concepts of circles. For instance theorems, principles, definitions, and properties of circles. How aspirants should prepare to score well in CAT quantitative ability easily.

Introduction of Circles

A circle is a closed two-dimensional figure. Which is formed by a set of points that lie on the same plane. Even a circle is at an equal distance from a certain point. That point is called the centre of the circle. Hence, the distance of any point on the circle from its centre is considered the radius of the circle.

Important terms to understand Circles for CAT exams.

These terms will help you to understand the basics of the circle. To solve questions accurately basic terms are important. So, have a look below:

1. Arc:

An arc is just a part of a circle. Degrees are measured by an arc. An arc is measured in degrees of the angle it makes with the centre.

Properties of the Arc of circles are :

(i) The measure of an Arc

1. A semi-circle is also an arc with a measure of 180 degrees.
2. The measure of a minor arc is the measure of the smaller angle it makes with the centre. Let us consider a circle where BC is the minor arc. With measure 45 degrees.
3. The measure of a major arc BAC is 360-(a measure of corresponding minor arc) =360-m(Arc BDC) = 360– 45= 315 degrees

(ii) Intercepted Angle

An angle with vertex (A) as one point on the arc apart from its end-points( B and C here) and forming a triangle BAC so the sides BA and CA make an angle CAB. Arc BAC inscribe the angle CAB

(iii) Intercepted Arc

An arc is intercepted by an angle when the sides that make the angle contain an endpoint of the arc, and the arc lies in the interior area of the angle, except for its endpoints. Arc DB and arc CA are intercepted by the COA.

2. Tangent

A tangent is a line that touches a circle at only one point. Therefore, the tangent is perpendicular to the radius at the point of contact.

In the above diagram, the line containing the points B and C is a tangent to the circle. Hence, It touches the circle at point B. So it is perpendicular to the radius OB. BC is perpendicular to OB.

3. Chord

A chord is also a line segment with both endpoints on the circle, but it may not pass through the centre of the circle.

4. Secant

A secant is a line which intersects the circle. Intersect circle in two distinct points. See the below figure for a definition of chord and segment

Circumference and Area formula for Circles in CAT exam.

Circumference:

The circumference of a circle is the distance around a circle.

The formula for the circumference of a circle is C = πd (π =3.142)

Or C = 2πr, where C is the circumference, d is the diameter and r is the radius.

Area

The area of a circle is the region enclosed by the circle.

It is given through the formula: A = πr² where A is the area and r is the radius.

Types of circles for CAT exam:

Concentric Circles: Circles lying in the same plane with a common centre. Hence these circles are considered Concentric circles.

Tangent Circles: Circles lying in the same plane and having only one point in common. Hence these circles are considered tangent circles.

One and only one circle passes through three given non-collinear points. An infinite number of circles pass through two given points.

Properties important in circles for CAT exam:

Properties of Chords, Tangents, and Secants are important from an exam point of view. Some direct and indirect questions could arise from these properties. Therefore, Have a look below to understand these properties in detail.

1. Chords

PROPERTY 1: A line from the centre of a circle perpendicular to a chord of the circle bisects the chord into two equal parts. Conversely, it can consider that the line segment joins the centre of the circle. So the midpoint of any chord makes a right angle with the chord.

If OM =AB, then AM=MB and If AM=MB, then OM=AB

PROPERTY 2: Chords of a circle or congruent circles that are equal, are equidistant from the centre of the circle. Conversely, two chords inside a circle or two congruent circles that are at the same distance from the centre of the circle, make the same angle with the line drawn from

the centre is equal.

If AB=CD, then OM=ON and if OM=ON, then AB=CD

PROPERTY 3: “Equal chords subtend equal angles at the centre”. This is true for one circle or a group of congruent circles. Conversely, chords that subtend equal angles at the centre of one circle or congruent circles, are equal in length.

If AM=CD, then m ∠COD= m ∠AOB and If m ∠COD= m ∠COD, then AB=CD

2. Tangent

PROPERTY 4: Any tangent to a circle and the radius through the point of contact are perpendicular to each other. If O is the centre of the circle, A is the point of contact of the tangent X, then OA 丄 X

Take any point on the circle, there is only one line passing through that point that is the tangent to that circle. From any point outside the circle, precisely one could draw two tangents onto that circle. One could not draw a tangent from any point in the circle.

PROPERTY 5: The lengths of two tangents to the circle, from any external point, are equal. One could draw a tangent from point C lying outside the circle. Touching the circle at points A and B, then AC=BC

5.1) In two tangent circles, the point of contact lies on the straight line through the centre of both circles.

5.2) In two tangent circles, the distance between the centre of both circles = the sum of their radii.

5.3) If any two circles touch each other internally at one point, the distance between the centre = the difference of the radii.

Distance between centre AB = | AC-BC | where B is the point of contact.

3. Angle subtended by a diameter

PROPERTY 6: ( i ) The diameter subtends an obtuse angle at any point E lying inside of the circle AEB>900

(ii) The diameter of a circle subtends an acute angle at any point E in the exterior of the circle AEB<900

(iii) The diameter of a circle subtends a right angle at any point lying on the circle. AEB=900

Also, if a line segment subtends a right angle at a point on the circle, then the line segment is the diameter of the circle

PROPERTY 7: Angles inscribed by one arc on any point on the circle are equal ∠AOB=∠ACB as they are inscribed by the same arc AB. O and C are two distinct points on the circle.

PROPERTY 8: Equal arcs of a circle make equal chords. Conversely, equal chords of a circle make arcs equal to each other.

PROPERTY 9: Inscribed angle theorem: The measure of an angle inscribed by an arc (at any point on the circle) is half the measure of the arc.

PROPERTY 10: The sum of opposite angles of a cyclic quadrilateral is always 180 degree

If a and b are opposite angles, a +b=180

4. Some more Angle subtended by a diameter properties

PROPERTY 11: “If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the segment, then the four points lie on the same circle.”

Points A, B, C, and D lie on one circle; i.e. they are concyclic points.

PROPERTY 12: If two secants intersect outside of the circle, the angle that they intersect at is equal to half of the difference in the length of the arcs intercepted by them on the circle.

PROPERTY 13: In case two secants intersect inside the circle, the angle they intersect at is equal to half the sum of the measures

of the arcs intercepted by them.

AOC=1/2[m (arc AC) +m (arc BD)] Also, AO x BO = CO x DO

Note: – O need not be the origin

PROPERTY 15: If a tangent and a secant intersect outside the circle. The angle of intersection is half the difference in length. Half of intercept an arc.

5. Common Tangents

PROPERTY 16: The two circles that have centres A and B. Where QP and SR are two direct common tangents. DC and FE are two transverse common tangents of both circles. One could draw two circles only with two of each direct common and transverse

Where r1 and r2 are the radii of the two circles

Length of Direct common tangent PQ = √ ((AB)2 – (r1– r2 )2 )

Length of the transverse common tangent CD = √ ((AB)2 – (r1+ r2 )2 )

Also read: How to Prepare for CAT

Some practice questions of circles for CAT exams.

Example 1: Five concentric circles are drawn whose circumferences are in the ratio 2:3:7:9:13. If a line is drawn joining the centre to the outermost point on the circle, what is the ratio of the length of the line to the radius of the second innermost circle?

Solution: Since the circles are concentric, the line joining the centre of the circle to the point on the outermost circle will be the radius of the outermost circle.

Now, Let the length of the radius of the inner circle be 2x.

Thus, the length of the radius of the second circle would be 3x.

Similarly, the length of the radius of the outermost circle would be 13x.

Thus required ratio = 13x/2x = 13/2

Example 2: A palm tree swings with the tree in such a manner that the angle covered by its tree is 14 degrees. If the topmost portion of the tree covers a distance of 33 meters, find the length of the tree.

Solution: Let the length of the tree be OA. As per the question, AB is the distance. Thus the length of arc AB = 33 meters. Also, ∠AOB = 14 degrees.

We know that for a circle with radius r (OA = length of the tree) and angle θ at the centre. 2πrθ/360

In our case, r= OA and θ = 14 degrees

Thus, 2 * (22/7) * r * (14/360) = 33 meters

Solving, we get r = 135 meters

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