Geometry for CAT is one of the most frequently asked topics in the CAT Quantitative Aptitude Section. Over 20% of the CAT paper is asked from this topic alone. Geometry includes topics such as Triangles, Lines and Angles, Quadrilaterals, Circles, and so on. Several questions do not require the knowledge of complex geometrical theorems and can be solved directly by using formulas or knowledge of basics. Check CAT 2021 Syllabus. 

Geometry deals with measures and properties of points, lines, surfaces, and solids, and the topics from where questions can be asked in CAT Question Paper are Lines and their properties, Polygons & Circles and their properties,  Triangles & Quadrilaterals and their properties, etc. Read the article to know more about CAT Geometry important topics and formulas, basic theorems, and solved sample questions to prepare for CAT 2021. 

CAT Geometry: Angles and their properties

The major types of angles are as follows: 

Quick Links:

CAT Geometry: Polygons and their Properties

• Any closed plane figure with n number of sides is known as polygon
• If all the sides and the angles of this polygon are equivalent, then it is called a regular polygon. 
• Polygons can be convex or concave. 
• A convex polygon is a simple polygon that has the following features:
  • Every internal angle is at most 180°.
  • Every line segment between the two vertices of the polygon does not go outside the polygon (i.e., it remains inside or on the boundary of the polygon). 
  • Every triangle is strictly a convex polygon
• A polygon is named after the number of sides it has: 

Properties of a Polygon: 

• Interior angle + Exterior angle = 180°
• The number of diagonals in an n-sided polygon = n(n – 3)/2
• The sum of all the exterior angles of any polygon = 360°
• The measure of each exterior angle of a regular polygon = 360°/n
• The ratio of the sides of a polygon to the diagonals of a polygon is 2:(n – 3)
• The ratio of the interior angle of a regular polygon to its exterior angle is (n – 2):2
• The sum total of all the interior angles of any polygon = (2n - 4) × 90°
• So, each interior angle in a regular polygon (2n − 4)90/n°
• Two polygons are similar if: 
  • Their corresponding angles are equal
  • Their corresponding sides are proportional

Quick Links:

CAT Geometry: Triangles and their Properties

Properties of a triangle

• The sum of all the angles of a triangle = 180°
• The sum of lengths of the two sides > length of the third side
• The difference of any two sides of any triangle < length of the third side
• The area of any triangle can be found by using this formula: 
• Area of any triangle = 1/2 × base × perpendicular to base from the opposite vertex.

Types of Triangles

• Scalene Triangle: A triangle whose all sides are of different lengths is a scalene triangle.
• Isosceles Triangle: A triangle whose two sides are of equal length is called an isosceles triangle.
• Equilateral Triangle: A triangle with all sides of equal length is called an equilateral triangle.
• Right-angled Triangle: A triangle with one angle of 90° is called a right-angled triangle
• Obtuse-angled Triangle: If one of the angles of the triangle is more than 90°, then the triangle is known as an obtuse-angled triangle.
• Acute-angled Triangle: If all the angles of the triangle are less than 90°, then the triangle is known as an acute-angled triangle.
• Isosceles Right-angled Triangle: A right-angled triangle, whose two sides containing the right angle are equal in length, is an isosceles right triangle.
• Pythagoras Theorem: Pythagoras theorem is applicable in the case of a right-angled triangle. It says that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Square of (Hypotenuse) = Square of (Base) + Square of Perpendicular)

Important Theorems for Triangles

• Basic Proportionality Theorem (BPT): Any line parallel to one side of a triangle divides the other two sides proportionally. So, if DE is drawn parallel to BC, then it would divide sides AB and AC proportionally. 
• Mid-point theorem: Any line joining the mid-points of two adjacent sides of a triangle are joined by a line segment, then this segment is parallel to the third side. 
• Apollonius’ theorem: In a triangle, the sum of the squares of any two sides of a triangle is equal to twice the sum of the square of the median to the third side and square of half the third side. 
• Interior angle Bisector theorem: In a triangle, the angle bisector of an angle divides the opposite side to the angle in the ratio of the remaining two sides
• Exterior angle Bisector theorem: In a triangle, the angle bisector of any exterior angle of a triangle divides the side opposite to the external angle in the ratio of the remaining two sides

Rules for Two Triangles to be Congruent

• S - S - S: If in any two triangles, each side of one triangle is equal to a side of the other triangle, then the two triangles are congruent. 
• R - H - S: This rule is applicable only for right-angled triangles. If two right-angled triangles have their hypotenuse and one of the sides equal, then the triangles will be congruent.

Rules for Similarity of Triangles

• AAA similarity: If in two triangles, the corresponding angles are equal, then their corresponding sides will also be proportional (i.e., in the same ratio). Therefore, the two triangles are similar.
• SSS similarity: If the corresponding sides of two triangles are proportional (i.e., in the same ratio), their corresponding angles will also be equal and so the triangles are similar. 
• SAS similarity: If one angle of a triangle is equal to one angle of the other and the sides including these angles are proportional, then the triangles are similar. 
• The ratio of the areas of the two similar triangles is equal to the ratio of the squares of their corresponding sides.
• If a perpendicular is drawn from the vertex of the right angle of a right-angled triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.

Quick Links:

CAT Geometry: Quadrilaterals and their Properties

Types of Quadrilateral

• Parallelogram: A parallelogram is a quadrilateral when its opposite sides are equal and parallel. The diagonals of a parallelogram bisect each other.
• Rectangle: A rectangle is a quadrilateral when its opposite sides are equal and each internal angle equals 90°. The diagonals of a rectangle are equal and bisect each other. 
• Square: A square is a quadrilateral when all its sides are equal and each internal angle is 90°. The diagonals of a square bisect each other at right angles (90°). 
• Rhombus: A rhombus is a quadrilateral when all sides are equal. The diagonals of a rhombus bisects each other at right angles (90°)
• Trapezium: A trapezium is a quadrilateral in which only one pair of the opposite sides is parallel. 
• Kite: Kite is a quadrilateral when two pairs of adjacent sides are equal and the diagonals bisect each other at right angles (90°).
• Area of Shaded Paths

Case 1: When a pathway is made outside a rectangle having length = l and breadth = b

• ABCD is a rectangle with length = l and breadth = b, the shaded region represents a pathway of uniform width = W
• Area of the shaded region/pathway = 2w (l + b - 2w)

Case 2: When a pathway is made inside a rectangle having length = l and breadth = b

• ABCD is a rectangle with length = l and breadth = b, the shaded region represents a pathway of uniform width = w
• Area of the shaded region/pathway = 2w (l + b + 2w)

Case 3: When two pathways are drawn parallel to the length and breadth of a rectangle having length = l and breadth = b

• ABCD is a rectangle with length = l and breadth = b, the shaded region represents two pathways of a uniform width = w
• Area of the shaded region/pathway = W (l + b - w)

Quick Links:

• How to Prepare for CAT Seating Arrangement?
• How to Prepare for CAT Logical Reasoning?
• How to Prepare for CAT Venn Diagrams?

---

CAT Geometry Sample Questions

Ques: a, b, c are integers that are the sides of an obtuse-angled triangle. If ab = 4, find c?

1. 2
2. 3
3. 1
4. Various values are possible for c

Solution: 3

Ques: How many isosceles triangles with integer sides are possible such that sum of two of the side is 12?

1. 11
2. 6
3. 17
4. 23

Solution: 3

Ques: If the sides of a triangle are 6, 10, and z the find out that for what value of z is the area of the △ maximum?

1. 8 cms
2. 9 cms
3. 12 cms
4. None of these

Solution: 4

Ques: Perimeter of a △ with integer sides is equal to 15. How many such triangles are possible?

1. 7
2. 6
3. 8
4. 5

Solution: 1

Ques: △ABC has integer sides a, b, c such that ab = 12. So how many such triangles are possible?

1. 8
2. 6
3. 9
4. 12

Solution: 3

Ques: Let us consider a right-angled triangle with an inradius 2 cm and a circumradius of 7 cm. What is the area of the triangle?

1. 32 sq cms
2. 31.5 sq cms
3. 32.5 sq cms
4. 33 sq cms

Solution: 1

Ques. If each interior angle of a regular polygon is 135°, find the number of diagonals.

1. 54
2. 48
3. 20
4. None of these

Ques. The largest angle of a triangle of sides 7 cm, 5 cm, and 3 cm is:

1. 45°
2. 60°
3. 90°
4. 120°

Ques. The three sides of a triangle measure 6 cm, 8 cm, and 10 cm, respectively. A rectangle equal in area to the triangle has a length of 8 cm. The perimeter of the rectangle is:

1. 11 cm
2. 22 cm
3. 16 cm
4. None of these

Quick Links: