A Complete Guide For Geometry Basics For CAT
In this article, we are going to learn about the properties of triangles and geometry basics for CAT. Compiled by our experts at IIM Skills online CAT coaching
Contents of the article for geometry basics for CAT:
 Theory
 Practice questions for geometry basics for CAT
 Syllabus for CAT
Geometry is an important topic of quantitative aptitude in not only CAT but many other competitive exams. In this post, we are going to discuss various properties of triangles and geometry basics for CAT. It is important to be strong in geometry basics for CAT since this is the basis of a good score in geometry in the Quantitative aptitude section of CAT.
Theory in geometry basics for CAT
THE TRIANGLE
The triangle is a very basic and the simplest closed 2dimensional figure. We come across triangles in every class throughout school. Every year, there was something new to learn about triangles. It makes us wonder the simplest shapes could be so complex. Triangle is a central part of the geometry portion of any class. That’s because a triangle helps understand the basics of all geometry. All types of polygons with any number of sides can be divided and represented as a group of triangles.
A triangle is a polygon having 3 sides and 3 vertices. A triangle is drawn by joining 3 points that must be noncollinear, otherwise, it would become a straight line. There are 3 internal angles whose sum total is 180 degrees, and the sum of exterior angles is 360 degrees. There are different types of triangles. They are classified on the basis of the measurement of sides and angles. There is also a comparison between two triangles like congruency, similarity, etc.
Since the triangle was such a big part of geometry, it is a measure of how well the student would be in geometry. This is also the reason why triangles and geometry are so important in Geometry basics for CAT.
Some other properties of the triangles in geometry basics for CAT are
 The sum of the length of any two sides of a triangle is always greater than the third side i.e. AB + BC > AC and the difference between the length of two sides is always less than the third side,  AB – BC  < AC.
 The value of an exterior angle is the sum of two opposite interior angles.
 The side opposite of the biggest internal angle is the longest side of the triangle and vice versa.
Classification of Triangles
Triangles can be divided into two types
Based on the length of the side
Equilateral Triangle
In the equilateral triangle, the length of all three sides is equal. Thus, all three angles are also equal i.e. 60o
Area of equilateral triangle = √3/4*(side)2
where ‘side’ is the length of the side.
Isosceles Triangle
In this triangle, the length of the two sides is equal and one is different.
Also, the angles corresponding to these sides are also equal.
AREA = ½ x base x height
Based on the measure of angle
Scalene Triangle
In a scalene triangle, all the sides measure different from each other in length and for the same reason, the angles are also varying.
Acute angle triangle
In an acute triangle, all the angles of the triangle are less than 90. An equilateral triangle is an acute triangle since all its angles are <90. An isosceles and scalene triangle can also be an acute triangle.
Obtuse Angle Triangle
In an obtuseangled triangle, one of the angles is greater than 90. There cannot exist two obtuse angles in one triangle as the sum of all angles is 180. Therefore, in the obtuse triangle, one angle measures>90, and the other two are acute.
Right Angle triangle
A rightangled triangle is the one in which one angle measures 90 and the other two angles are acute angles and can be equal. This condition between the sides and angles of a right triangle is the basis for trigonometry.
IMPORTANT THEOREMS
These were the types of the triangle on the basis of sides and angles in geometry basics for CAT. Now we are going to learn about some formulae. These will help in the various scenarios in finding the length of sides or angles in the geometry questions.
For example, to find the area of a triangle, there is more than one method. Ways to find the area of a triangle are more than any other polygon. Let’s learn about some of them.
Heron’s Formula:
The most basic method for calculating the area of a triangle is
Area of triangle = ½ x base x height
But sometimes, when the height is not given but you know the lengths of all sides, we have a direct formula to find out the area of a triangle in that case. You do not need to do calculations to find the height before you find the area.
Let a, b, and c, be the length of the sides of the triangle then,
Area = (s*(sa)*(sb)*(sc))1/2
where, s = (a+b+c)/2
‘s’ is also called the semiperimeter because it is half the value of the perimeter.
The area through Trigonometry:
We can also use trigonometry equations to find out the area of a triangle if we have lengths of two sides and measure of one angle.
Area of triangle ABC = ½ bc x sinA = ½ ab x sinC = ½ ac x sin B
Determinant method:
The determinant method makes use of coordinate geometry to determine the area of a triangle. Therefore,
= area
where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the vertices of the triangle
There are a few other ways of determining the area of the triangle:
 Area of Triangle = abc/4r, where r is the circumradius. Circumradius is the radius of the circle that has vertices of the triangle as points on the circle
 Area of Triangle = r*s, where r = inradius and s = semiperimeter. Inradius is the radius of the circle that is drawn with the sides of the triangle as tangents to the circle
Now we will move forward to discussing altitudes, medians, and perpendicular bisector. Many students have confusion between each of them so we will clarify all of them here. What is the definition of each and what do we call their point of intersection.
Altitudes and orthocentre
An altitude is a line segment that passes through any vertex and cuts at a right angle with the side opposite to this vertex.
Here, AD, CF, and BE are altitudes of a triangle ABC.
The orthocenter is the point of intersection of all the three altitudes of the triangle. The orthocenter may lie either inside or outside of the triangle.
Here, O is the orthocenter.
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Perpendicular Bisector and Circumcenter
A line segment that passes through any vertex of the triangle to the middle point of the opposite side and cuts it at a right angle with it is called the perpendicular bisector.
In this figure, AD, CE, and BF are perpendicular bisectors.
The circumcenter is the intersection of all three perpendicular bisectors. it is also the center of the circumcircle.
Here, G is the circumcenter.
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Median and Centroid
The line segment that joins any vertex of a triangle with the midpoint of the opposite side is called the median. It divides the opposite side into two equal halves.
Here, QU, PT, and SR are medians of ABC.
The centroid is the point of intersection of the three medians of a triangle. The centroid divides the medians in a 2:1 ratio of lengths.
Here, V is the centroid.
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Angle Bisector and incentre
The line segment that bisects the angle into two angles of equal measure of the vertex from which it is drawn is called the angle bisector.
The incentre is the intersection of the three angle bisectors. It is also the center of the incircle of the triangle.
Here, I is the incentre.
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Congruency and Similarity
Congruency and similarity of triangles is an important concept in geometry basics for CAT. Here are some basics about the two concepts. A comparison is necessary because some candidates confuse the two concepts.
Congruency  Similarity  
Definition  Two triangles are said to be congruent if they have the same size and shape. All interior angles and corresponding sides measure the same.  Two triangles are said to be similar when all corresponding angles of the first triangle and the second triangle are equal and the lengths of corresponding sides in both the triangles are in the same ratio. 
Rule 1: SAS (side angle side)  Two triangles are congruent if two sides of a triangle are equal to corresponding sides of another triangle and the angle between them is also equal.  Two triangles are called similar when two corresponding sides in the two triangles are in the same proportion as each other and the corresponding angles are also equal. 
Rule 2: SSS (Side Side Side)  Two triangles are SSS congruent if all three sides in one triangle are equal to the corresponding sides in another triangle.  If three sides of the first triangle are in the same proportion with the corresponding three sides of the second triangle then, they are said to be SSS similar triangles.

Rule 3: AAS/ AA (Angle Angle Side)  If two corresponding angles are equal and one nonincluded corresponding side is equal in length in both triangles, then they are said to be AAS congruent.
 Two triangles are said to be AA similar if two pairs of angles equal 
Rule 4: ASA (Angle Side Angle)  This rule is only for congruency. It states that if the corresponding two angles and the included side in between them in both the triangles are equal, they are congruent.
 There is no ASA rule for the similarity of triangles 
Some important theorems:
There are countless theorems and formulae that we studied in mathematics. Especially in geometry. Many of these theorems were connected to the triangle because the triangle is the most basic twodimensional shape. Let us learn about a few of them.
Angle Bisector Theorem:
It states that the angle bisector of any angle inside the triangle divides the side opposite to it into the ratio of the length of the sides making the angle. To take an example, in a triangle ABC, let AD be the angle bisector of angle A and AD divide the side BC in the ratio m:n, then,
This theorem is valid for both the interior and exterior angles of the triangle. The abovegiven figure is for the angle bisector bisecting the interior angle. For exterior angle,
For exterior angle, in triangle ABC, let D be a point extending the line CB, such that AD be the angle bisector bisecting angle CAD of this triangle externally. Let AD divide the side BC in n:m ratio then,
The Pythagoras Theorem:
Pythagoras theorem is one of the most wellknown theorems in geometry basics for CAT. It states that in the case of a rightangled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The hypotenuse is the side opposite to right angle i.e. if there’s a triangle with a and b be the length of the sides containing the right angle and c be the length of the hypotenuse then,
This theorem is one of the most widely used theorems in geometry and related fields.
Apollonius’s Theorem:
The sum of the squares of any two sides of a triangle equals twice the sum of the square of half the third side and the median of the third side. In a triangle ABC, where AM is the median of that triangle such that BM=MC; then,
Midpoint Theorem:
According to the midpoint theorem, In a triangle, the line that joins the midpoint of the two sides is parallel to the third side and is half of it. Let there be a triangle ATV; and midpoints R and S such that AR=RT and AS=SV; then,
Basic Proportionality Theorem:
Basic Proportionality theorem states that inside a triangle if a line segment is drawn parallel to one side to intersect the other two sides, the other two sides get divided in the same proportion. Let there be a triangle ABC and DE be the line parallel to BC then,
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Now that we have read the necessary theory in geometry basics for CAT, let’s look at some practice questions to get an idea of the basics. As a rule, you should do as many practice questions as your timetable allows to get a hold of the geometry basics for CAT.
QUESTION: Given sides of a triangle ABC is 6, 10, and x. Find x for which area of the triangle is maximum?
 8
 √19
 12√3
 √136
SOLUTION:
We can solve this problem with Heron’s formula but there is a simpler way
We know the length of sides of a triangle is 6, 10, and x.
=> Area = 1/2 * 6 * 10 * sin∠BAC.
The area is maximum when
sin∠BAC=1;
=>∠BAC = 90◦
x = √(100+36) = √136 (by Pythagoras theorem)
Here D is the correct answer.
You should also try to solve using the heron’s formula
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QUESTION: Perimeter of an △ABC is 15. All sides have integral lengths. How many triangles can be made like that?
 7
 9
 8
 4
SOLUTION:
The solution to this question is pretty simple. You just have to use the trial and error method to satisfy the condition that the sum of two sides should be greater than the third side.
Let us assume a ≤ b ≤ c.
When, a = 1, Possible sides of triangle 1, 7, 7
When, a = 2, possible sides of triangle 2, 6, 7
When, a = 3, possible sides of triangles 3, 6, 6 and 3, 5, 7
When, a = 4, possible sides of triangles 4, 4, 7 and 4, 5, 6
When, a = 5, possible sides of triangle 5, 5, 5
So a total of 7 triangles are possible.
Therefore, the answer is 7.
Option A is the correct answer.
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QUESTION: A triangle △ABC has a length of sides as integer a, b, c such that ac = 12. How many such triangles are possible?
 10
 8
 9
 12
SOLUTION:
This is again faster to solve with the trial and error method. Let’s begin with all the possible causes of sides ab of the triangle.
ac = 12
Lengths a,c can be one of these 1, 12 or 2, 6 or 3, 4
Possible triangles with the given condition
1, 12, 12
2 , 6 , 5; 2 , 6 , 6; 2 , 6 , 7
3 , 4 , 2; 3 , 4 , 6; 3 , 4 , 3; 3 , 4 , 4; 3 , 4 , 5;
Hence, the correct answer is 9 triangles.
Option C is the correct answer.
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QUESTION: ABCDE is a regular pentagon. O is a point inside the area of pentagon such that AOB forms an equilateral triangle(all sides equal). What is the value of ∠OEA?
 66
 62
 54
 75
SOLUTION:
Creating a new triangle will help in understanding.
Join OE and OD.
In a regular pentagon, all 5 sides are equal, hence the internal angle opposite to the sides are also equal.
Internal angle of regular pentagon = 108°
∠BAE = ∠CDE = 108°
=> ∠BAO = 60°
=> ∠OAE = 48°
OA = OB = AB since the triangle is equilateral.
And BA = EA in a regular pentagon.
=>Triangle AEO is isosceles as AO = EA.
Let ∠OEA = ∠EOA = a
In triangle AEO,
∠OAE + 2a = 180°
48° + 2a = 180°
2a = 132°, or a = 66°
=>∠OEA = 66°
Therefore, the correct answer is 66°
Option A is the correct answer.
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QUESTION: Consider a rightangled triangle with an inradius 2 cm and a circumradius of 7 cm. What is the area of the triangle?
 32 sq cm
 33.5 sq cm
 32.5 sq cm
 31 sq cm
SOLUTION:
Given,
r = 2 and
R = 7 (Half of length of hypotenuse)
Hypotenuse = 14
r = (a + b – h)/2
=> 2 = (a + b – 14)/2
=> a + b – 14 = 4
=> a + b = 18
=> a2 + ba2 = 142
=> a2 + (18 – a)2 = 142
=> a2 + 324 + a2 – 36a = 196
=> 2a2 – 36a + 128 = 0
=> a2 – 18a + 64 = 0
By finding the roots of a quadratic equation,
The 2 roots to this equation will effectively be a, (18 – a).
Product of the roots = 64.
=> Area = 1/2 * product of roots
= 32 sq. cms
Therefore, the answer is 32 sq. cms
Option A is the correct answer.
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QUESTION: A triangle ABC has a perimeter of 6 + 2√3 . One of the interior angles of ABC is equal to the exterior angle of a regular hexagon. Another angle of the triangle is equal to the exterior angle of a regular 12sided polygon. Find area of the triangle.
 2√3
 √3
 √3/2
 3
SOLUTION:
Given, Perimeter of the triangle = 6 + 2√3
Given, one of the angles in the triangle is equal to the exterior angle of a regular hexagon ie. 60°
And the second angle is equal to the exterior angle of a regular polygon of 12 sides ie. 30°.
From this, we can calculate that the third angle in the triangle is 90°.
In a triangle with angles 60,30,90, the sides are in the ratio √3a, a, and 2a.
=> Perimeter is sum of all sides = a(3+ √3)= 6 + 2√3 .
=> a = (6 + 2√3)/(3+ √3) = 2.
=> the sides of the triangle are 2√3, 2 and 4.
Area of a Right Triangle = 1/2 X Product of smaller sides = 1/2 X 2 X 2√3 = 2√3 .
Therefore, the answer is 2√3.
Option A is the correct answer.
Syllabus for CAT
Syllabus for Quantitative ability
 Number Systems
 Mean median mode (statistics)
 Averages
 Percentages
 LCM and HCF
 Time and Work
 Ratio and Proportion
 Profit, Loss, and Discount
 Speed, Time, and Distance
 Quadratic Equations & Linear Equations
 Complex Numbers
 Simple and Compound Interest
 Logarithms
 Sequences and Series
 Linear equations
 Inequalities
 Probability
 Surds and Indices
 Set Theory & Function
 Permutation and Combination
 Mixtures and Alligations
 Trigonometry
 Coordinate Geometry
 Geometry
 Mensuration
Syllabus for DILR
 Tables
 CodingDecoding
 Letter Series
 Symbol Series
 Symbol based Logic
 Number & Alphabet Analogies
 Odd one out
 Direction Sense
 Blood Relations and Family Tree
 Cryptarithmetic (Verbal Arithmetic)
 Inequalities and Conclusions – Coded Inequalities
 Data Sufficiency
 Approximation of Values
 Caselets
 Bar Graphs
 Line Charts
 Column Graphs
 Venn Diagrams
 Pie Chart
 Calendars
 Number and Letter Series
 Clocks
 Cubes
 Seating Arrangement
 Binary Logic
 Logical Matching
 Logical Sequence
 Syllogism
Syllabus for VARC
 English Usage or Grammar
 Vocabulary (Synonyms/ Antonyms)
 Fill in the blanks
 Sentence Correction
 Jumbled Paragraph
 MeaningUsage Match
 Analogies or Reverse Analogies
 Summary Questions
 Verbal Reasoning
 FactsInferencesJudgments
 Reading Comprehension
 Categories of Passages
 Writing Styles
 Tone of Writing
 Types of Questions
 Reading Skills
 The Articles – A, An, The
 Grammar rules
 Parts of Speech in English
 Sentence Construction in English
 Punctuations
 Modifiers
 SubjectVerb Agreement
 Verbs Tenses
 Word Usage
 Verbal Reasoning
 Logical Deduction
 Statements & Assumptions
 Courses of Action
 Para jumbles (Basic Rules, Extra tips)
 Para Completion
 Sentence Exclusion
 Fact Inference Judgment
 Syllogism
 Basic Assumption And Inference
 Paragraph Summary
 Method of Reasoning and Boldfaced
 Flawed and Paradox
 Parallel, Further Application, Evaluate
 Fallacies
 Strong Weak Arguments
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