Cat 2015 geometry concept session 1

Session for CAT 2015
Alosies George
IIM Calcutta
director@georgeprep.com
+91- 9985-372-371
7/5/2015 www.georgeprep.com
1
Geometry Session 1

2
Introduction

What it takes to solve a question from
geometry?
3

4
TrianglesBasic Properties
Geometric Points in a
Triangle
Congruence and
Similarity
Equilateral and
isosceles triangle
Area of a triangle
Basic theorems
Polygons
General properties of Convex
and concave polygon
properties
Regular hexagon and octagon
Quadrilaterals
• Cyclic quadrilateral
• Parallelogram
• Rectangle
• Rhombus
• Square
• Trapezium
3 D figures
• Cube
• Cuboid
• Cone
• Cylinder
• Sphere
• Hemisphere
• Pyramid
• Fulcrum
Circle
General properties

5
Building Blocks of
Geometry

Building blocks of Geometry
6
AB
C
Points, lines and planes
Collinear and Coplanar points
Collinear Points – points on same line
Coplanar points – points on the same plane

Angles
An angle consists of two rays with a common end point (or, initial point).
 Each ray is a side of the angle.
 The common endpoint is called the vertex of the angle.
7Building blocks of Geometry

Classification of angles
 Acute
 Right
 Obtuse
 Reflex
Reflex

Supplementary angles and complementary angles

10
Vertically opposite angles Linear Pair

11
Parallel lines and transversal

14
In the figure, AB is parallel to CD, if Angle
ABE = 110˚ and angle BED = 25 ˚ find
angle CDE ( in degrees)
1. 115
2. 125
3. 135
4. 55
5. None of these
A
E
D
B
C
Answer: Option 3

15
AB is parallel to CD. Angle QPB = 105 degrees and
Angle QRC = 35 degrees Find the measure of
angle PQR ( in degrees)
1. 115
2. 125
3. 120
4. 55
5. None of these
A
D
B
C
Answer: Option 3
R
Q
P

16
Triangles – Basic Properties

Triangles
 Basic Properties
 Geometric Points in a Triangle
 Congruence and Similarity
 Equilateral and isosceles triangles
 Area of a triangle
 Basic theorems
17

Triangles
Basic Properties
• The sum of three internal angles = 180
• External angles?
• Exterior angle is equal to sum of interior opposite
Side related properties
• Sum of any two sides is greater than the third side
• Difference of any two sides is lesser than the third side
• Side opposite the greater angle is greater
Classification
• On the basis of angles
• On the basis of sides
18

Problem
19
The two sides of a triangle are 9cm and 20cm. If the magnitude of the third side is an odd number,
what is the sum of all the possible lengths of the third side?
1. 120
2. 160
3. 180
4. None of these

Problem
20
How many non-congruent triangles can be formed with a perimeter of 9 units with each side having
integer unit lengths?
1) 3
2) 2
3) 4
4) None of these

Sine Rule
 If two sides of a triangle and an angle opposite one of the two sides is given
can you find out the angle opposite the other side?
21
a/Sin A = b/ sin B = c/ Sin C =
2R

Cosine Rule
 If three sides of a triangle are given can you find the three angles of the
triangle?
22
a2 = b2+c2 – 2bc.cosA
b2 = a2+c2 – 2ac.cosB
c2 = a2+b2 – 2ca.cosC

Acute, Right and Obtuse on the basis of sides
Can you find out whether a triangle is acute if you know the dimensions of its sides?
23
a2 = b2+c2 – 2bc.cosA
b2 = a2+c2 – 2ac.cosB
c2 = a2+b2 – 2ab.cosC

Consider obtuse-angled triangles with sides 8 cm, 15 cm and x cm. If x is an integer
then how many such triangles exist?
1. 5
2. 21
3. 10
4. 15
5. 146
- CAT 2008
Problem

25
Triangles – Geometric
Points

Geometric points in a triangle
Centroid
- medians
- 2 triangles of equal area (6 triangles of equal area)
- 2:1 ratio
- Sum of squares of median/(sum of squares of sides)
= 3/4
26

Appolonius theorem
Centroid
- How can we find out the length of a median if the three sides of a triangle are given?
27
AB2 + AC2 = 2( AD2 + BD2)

Orthocenter
- Altitudes
- Position of Orthocenter and the nature of the
triangle
28

Orthocenter
- If three sides of a triangle are given can you find the foot of the perpendicular from a
vertex.
29
b2 = a2+c2 – 2 BDxBC b2 = a2+c2 + 2 BDxBC

Circumcenter
- Perpendicular bisectors
- circumcircle
- Position of circumcenter and the nature of the
triangle
30Geometric points in a triangle

Incenter
- Angle bisector
- Internal angle bisector theorem
- Incircle
- Angle ADB= 90-(AngleA)/2
- Excenters

The line that connects the centroid (G), the orthocenter
(H), and the circumcenter (C) is called the Euler Line.
GH = 2GC.
Euler’s Line

Problem
Let PQR be a triangle and O be a point such that areas of triangles POR, POQ and ROQ are equal.
If PQR is a scalene triangle which of the following must be true?
1. O lies on the altitude to base QR
2. O lies on the median to base QR
3. O lies on the perpendicular bisector to base QR
4. O lies outside triangle PQR
33

Cat 2015 geometry concept session 1

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