# How To Solve Circles For CAT

**Circles for CAT forms a big part of the geometry topic in the quantitative section. Clear your basics with this article prepared by IIM Skills – online CAT coaching**

Circles and their concepts are an important topic of the geometry section of CAT quantitative ability. CAT asks many questions from circles in the quant section. Many questions in quant CAT that are not directly asking circles make use of some indirect concepts of circles.

Here we will learn important concepts of circles for CAT, theorems, principles, definitions, properties of circles, and how to prepare for CAT quantitative ability easily.

## Definition of Circle:

A circle is a closed two-dimensional figure formed by a set of points that lie on the same plane and are at an equal distance from a certain point. That point is called the center of the circle and the distance of any point on the circle from its center is called the radius of the circle.

**Terms related to a Circle:**

Let us talk about various concepts and definitions that come in circles for CAT.

### a) Diameter

The diameter is the distance between two opposite points on a circle. The diameter passes from the center of the circle. The diameter is double the length of the radius. All diameters have the same length. A circle can have many diameters since it is a line segment joining any two diametrically opposite points.

### b) Radius

Any line segment from any point on the circle to the center of the circle is the radius. The length of the radius is half of the diameter.

In a circle with center O and two points A and B on the circle, and are the radii of this circle. The radii of any circle are all equal in length.

### c) Semi-circle

One half of a circle cut along any diameter is a semi-circle.

### d)Arc

An arc is just any part of the circle. An arc is measured in degrees of the angle it makes with the center.

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### Important properties of Arcs:

(i) The measure of an Arc

- A semi-circle is also an arc with a measure of 180 degrees.
- The measure of a minor arc is the measure of the smaller angle it makes with the center. Let us consider a circle where BC is the minor arc. With measure 45 degrees.
- The measure of a major arc BAC is 360-(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. This Angle CAB is inscribed by the arc BAC.

(iii) Intercepted Arc

An arc is intercepted by an angle when 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.

### e) Tangent

Any line that touches the circle only once is called a tangent to that circle. All tangents are perpendicular to the radius at the point of contact.

In the above figure, the line with points B and C is a tangent to the circle. The tangent touches the circle at point B and is perpendicular to the radius OB ie. BC is perpendicular to OB.

### f) Chord

A chord is a line segment with both ends lying on the circle, but it does not have to pass from the center of the circle.

### g) Secant

It is a line that intersects the circle at two different points. A secant is just an extended form of a chord.

### h) Circumference

The circumference of the circle is the length of the perimeter of a circle.

The formula for the circumference of a circle is

**Circumference of circle C = ?d = 2?r** (? =3.142)

where C = circumference, d = diameter, and r = radius.

### i) Area

The area of a circle is the area inside the boundary of the circle.

The formula for the area of the circle is:

**Area of a circle, A = ?r****2**** **

where A = area and r = radius.

**Types of circles:**

These are various types of circles you need to know for circles for CAT.

### (i) Concentric Circles:

Circles having the same center and lying on the same plane are called concentric circles.

### (ii) Tangent Circles:

Circles that lie in the same plane and have only one point in common are called tangent circles

Only one circle can pass through any 3 non-collinear points.

An infinite number of circles can be drawn passing through any 2 points.

**Properties Of circles: Chords, Tangents, and Secants:**

QUestions in quantitative ability CAT for circles for CAT can come either directly or indirectly on these properties.

#### PROPERTY 1: A line from the center of a circle perpendicular to a chord of the circle bisects the chord into two equal parts. Conversely, it can be said that the line segment joining the center of the circle and the midpoint of any chord makes a right angle with the chord.

#### PROPERTY 2: Chords of a circle or congruent circles that are equal, are equidistant from the center of the circle.

#### Conversely, two chords inside a circle or two congruent circles that are at the same distance from the center of the circle, that make the same angle with the line drawn from the center are equal.

#### PROPERTY 3: “Equal chords subtend equal angles at the center”. This is true for one circle or a group of congruent circles.

#### Conversely, chords that subtend equal angles at the center of one circle or congruent circles, are equal in length.

#### PROPERTY 4

**Tangent Perpendicularity Theorem:**

Any tangent to a circle and the radius through the point of contact are perpendicular to each other. If O is the center 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 two tangents can be drawn onto that circle.

No tangents can be drawn from any point inside the circle.

#### PROPERTY 5: The lengths of two tangents to the circle, from any external point, are equal.

If two tangents are drawn from a 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 centers of both circles.

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

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

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

#### PROPERTY 6: **Angle subtended by a diameter**

(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 arc 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

#### 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, 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 of the length of the arcs intercepted by them on the circle.

#### PROPERTY 13: If 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.

#### PROPERTY 15

If a tangent and a secant intersect outside the circle, the angle of intersection is half the difference of the length of the arc intercepted by them.

#### PROPERTY 16: **Common Tangents**

16.1)The two circles that have centers A and B, Where QP and SR are two direct common tangents and DC and FE are two transverse common tangents of both circles(Only two of each of direct common and transverse can be made with two circles).

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 )

These were all the properties you need to learn to score in circles for the CAT section in quant.

**Importance of Quantitative ability**

So why does CAT ask quant problems? And specifically questions on circles for CAT?

Today, all organizations need new employees to produce a specific level of quantitative ability in order to solve and deal with day-to-day issues. CAT is not the only competitive exam that tests the Quantitative ability of the candidate. Many competitive exams other than CAT analyze the candidate’s quantitative ability.

Now we will explain why a high score in Quantitative ability CAT is so vital for management schools as well as business organizations.

An aptitude test is designed to determine the likelihood of the success of the candidate in their career. In this way, the quantitative ability CAT score determines the odds of the success of a candidate in a management career.

The aptitude quiz is also normally used as a standard practice in various organizations to screen job applicants. Aptitude tests assist in measuring your strengths and weaknesses on different parameters. It is a means to test the analytical ability and awareness of the aspirant.

Quantitative ability analyses numerical and problem-solving skills. It forms a basic section in all competitive exams in India as well as many abroad, such as in CAT, CMAT, IIFT, XAT, MAT, GMAT, GRE, and so on.

An applicant with great Quantitative Aptitude is likely to identify and process numerical to perform elementary arithmetic methods and perform calculations. The applicant is likely to be fast in analyzing cases based on the provided information and shows an extraordinary level of concentration in solving difficulties.

A good score in the quantitative aptitude in CAT proves that you are well-equipped in the following areas:

- Mental sharpness: Quantitative ability CAT score is a measure of your mental sharpness. You will come across several expected and unexpected challenges as part of your daily work life. Successful business organizations need to be wary of other businesses in the same domain. They always need to be aware of every step of others.
- Problem-solving: Enterprises have several departments or divisions for different operations such as finance, strategy, legal, human resources, etc. There are many difficulties to encounter in the world of business every day. You need to be able to identify, think over, and find solutions quickly. A high Quantitative aptitude number shows that the applicant is has a talent for thinking critically and resolving problems.
- Analysis: One implication of proficiency in quantitative ability is reliable analytical abilities. Data Analysis forms the backbone of any business operations in this digital economy. It helps a firm decide on short-term and long-term plans that are essential for growth. An employee with high quantitative ability is appropriately suited for these duties.

Recruitment is not only about the candidate’s desires but also about what the company desires. This is why almost 80% of all the competitive exams in the world have some of the other variations of quantitative ability.

An inadequate worker could probably be the best use of the company’s resources and may be inadequate for the company’s goals. A better role will be good for the employee themselves as they will have more opportunities to grow.

In lieu of so lakhs of applications in CAT, the entrance exams need to be designed in such a way so they can differentiate between so many candidates.

**Why CAT?**

CAT or the Common Admission Test, is a necessary step for admissions into any management course in India. The format of the CAT is such that it tests the candidates in several different fields. The main purpose of the CAT or the common admission test is to predict the likelihood of how successful the candidates’ careers would be. Almost all management institutions in India shortlist and call applicants on the basis of their CAT scores.

CAT is a computer-based test. It has 3 different sections:

- Quantitative Ability (QA)
- Verbal Ability (VA) & Reading Comprehension (RC)
- Data Interpretation(DI) & Logical Reasoning (LR)

The Quantitative ability or simply quant(QA) has questions on elementary arithmetic, based upon 10 + 2 level. Questions come from various topics such as Number systems, Geometry, Algebra, circles for CAT, coordinate geometry, statistics, simple and compound interest, etc. The purpose is to examine the candidate’s ability to use basic math skills to find solutions.

The Verbal Ability (VA) has puzzles that test the candidates’ knowledge of the English language and systems of grammar. Verbal Ability questions analyze the candidate’s linguistic knowledge in the form of questions such as arranging paragraphs in proper order, determining the tone of the passage, and fill in the blank.

The DILR (Data Interpretation, Logical Reasoning) section has puzzles that come in different sets. There are usually 4-5 sets. Each set contains a group of data and has 3-4 questions. Data Interpretation questions check how the applicant makes use of the available data to make logical conclusions and solve problems.

The Common admission test was started by the IIMs to screen candidates seeking entry into their management colleges. Different IIMs make the papers every year. Over time, CAT scores began to be used by almost every management school including the ones in IITs(SJM school of management) and IISc.

The marking scheme in CAT is like this. Each correct answer awards 3 marks to the candidate. A wrong answer decreases 1 mark.

It is expected that more than 2.5 lakh students will register for CAT 2022. This is a lot of competition and a good percentile in CAT is required to get a seat in the most sought-after management institutions like the IIMs.

A high CAT percentile ensures a great learning atmosphere in any college you choose. You form a large and valuable corporate network, which is the envy of many. You get to sit and talk with the biggest industry leaders and prominent economists. You get a great start to your career. The topmost B-schools present great internship opportunities and even possess a 100% placement history.

### Why an MBA is important?

MBA is very important if you want to begin your management career. It has benefits not just for your career but also for your personality. Let’s discuss them one by one.

- It provides credibility: An MBA from a top business school provides you credibility in your circles. You are expected to uphold the standards of your university.
- It enhances your personality: As you act on projects with great partners, both inside the institute and outside, you develop confidence and realize your value and worth. You become a professional.
- Many skills are transferable skills that you learn from your team members at the college. Some of these skills are communication, time management, confidence, selling techniques, and so on.
- You can join any industry be it automobiles, banking, start-up, entertainment, arts, sports, or even politics.
- Strategic thinking is another major skill you get better at. This is a gift that is extremely valuable where you learn to examine the advantages and disadvantages of any decision.
- You grow and get better at communication. Management is all about communication. Communication of thoughts, ideas, plans, strategies, and so on. Effective communication is absolutely essential. It gets the job done. As a manager, you will have many learnings on different communication styles with different kinds of people.
- Discipline. To keep pace with your course, you need to attend many lectures, events, complete assignments within strict and tight deadlines, do projects, join in the study sessions. All the exercises require hard mental and physical labor and discipline.
- Time management. This follows from the previous point. You learn to manage your time effectively. You become productive and efficient. You waste less, perform more. You learn to utilize every chunk of time in a day.
- Broaden your perspective. MBA provides you with a window to the real world. You discover so much about the concerns in society. You learn how small pieces fit together to make a complex world.
- Besides all these skills, an MBA from a prime business institution provides you the best asset of your life in the form of a large and valuable business network. You meet numerous industry leaders and professors who are prominent economists and policymakers.
- An MBA sometimes can revive your stuck and shaky career. It happens oftentimes that a company does not promote some employees to a higher managerial role despite good performance, because they do not have an MBA degree.
- Thinking creatively and looking for novel solutions is another skill that many do not discuss when considering the benefits of an MBA program. Creative thinking is crucial in businesses like it in the arts and entertainment field.

Now you know why an MBA is so important for your career.

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