CAT Geometry is one of the important topics for CAT quants preparation. As the question is this topic has been asked frequently by the examiner in past. Geometry questions contain 20% weightage in the CAT exam. Geometry includes many topics. For example Triangles, Lines and Angles, Quadrilaterals, Circles, and so on.

CAT Geometry angles and properties.

There are 10 types of angles in geometry. Namely Complementary, Adjacent, Obtuse and Right Angle. Have a look below for more clarity.

1. Acute angle= The angle which is less than 90°.
2. Right angle= The angle which is equal to 90°.
3. Obtuse angle= The angle which is greater than 90° but less than 180°.
4. Straight Line= Angle which is equal to 180°.
5. Reflex Angle= Angle which is greater than 180° but less than 360°.
6. Complementary Angle= Two angles, whose sum is 90°, are complementary to each other.
7. Supplementary angle= Two angles, whose sum is 180°, are supplementary to each other.
8. Adjacent angles= Adjacent angles must have a common side.
9. Linear pair= One side must be common (e.g., OB), and these two angles must be supplementary.
10. Angle Bisector =Angle bisector is equidistant from the two sides of the angle, that is when a line segment divides an angle equally into two parts, it is said to be the angle bisector.

Types of triangles and their properties in CAT Geometry

Types of triangle

A triangle is a polygon with three edges and three vertices. It is formed by joining 3 non-collinear points in the 2-dimensional plane. And as the name itself signify the most elemental property of it i.e. the word triangle comes from joining Tri with an angle where tri means three thus it has 3 angles that sum up to 180^o where the 3 angles are the interior angles of the triangle given in the figure below.

And the sum of all the exterior angles is 360^o.

The other basic properties of triangles are

• The sum of two sides of a triangle is always greater than the third i.e. AB + BC > AC and the difference between the two sides is less than the third, AB – BC < AC.
• The measure of an exterior angle is the sum of two opposite interior angles.

Categorization of Triangles

Triangles can be divided into two types

1. Based on the length of a side.
2. Based on the measure of an angle.

Based on the length of side	Meaning	Based on the measure of angle	Meaning
Equilateral Triangle	In this type of triangle, the length of all three sides is the same and equivalent. Thus, the three angles are also equal i.e. 60o AREA = √3/4*a2, where a is the length of the side.	Acute angle triangle	In an acute triangle, all the angles of the triangle are less than 90o. An equilateral triangle is an acute triangle since all its angles are <90o. An isosceles and scalene triangle can also be an acute triangle.
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 equal. AREA = ½ base x height	Obtuse Angle Triangle	In an obtuse triangle, one angle measure greater than 90o. There cannot exist two obtuse angles in one triangle as the sum of all angles is 180o. Therefore, in the obtuse triangle, one angle measures>90o and the other two are acute. An equilateral triangle cannot be obtuse.
Scalene Triangle	In a scalene triangle, all the sides measure different from each other and for the same reason the angles are also contrasting.	Right Angle triangle	A right triangle is one in which one angle measures 90o and the other two angles are acute and can be equal. The relation between the sides and angles of a right triangle is the basis for trigonometry

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 between 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.

Some Important Theorems of Triangle in CAT Geometry

• 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 is joined by a line segment, and 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 the 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

Heron’s Formula:

The most elementary formula for calculating the area of a triangle is ½ base x height but at times it’s difficult to compute the height of the triangle. No worries, you don’t have to find out the height the length of all the three sides will work. Heron’s Formula evaluates are as follows

Let a, b, and c, be the length of three sides of the triangle then,

Area = (s*(s-a)*(s-b)*(s-c))^1/2,

Where, s = (a +b +c)/2

Polygons and its Properties in CAT Geometry.

• Any closed plane figure with n number of sides is known as a 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. For example :

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 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

Quadrilaterals and their Properties in CAT Geometry.

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°. So 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 bisect each other at right angles (90°)
• Trapezium: A trapezium is a quadrilateral in which only one pair of opposite sides is parallel.
• Kite: Kite is a quadrilateral where two pairs of adjacent sides are equal and the diagonals bisect each other at right angles (90°).
• Area of Shaded Paths

Some cases of Quadrilateral mentioned below:

1st Case: 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. Hence 2w (l + b - 2w)

2nd Case: 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. Hence 2w (l + b + 2w)

3rd Case: 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. Hence W (l + b - w)

Also read: How to Prepare for CAT 2022

Some Sample Questions

For example: Given sides of a triangle ABC is 6, 10, and x. Find x for which area of the triangle is maximum.

1. 8
2. √19
3. 12√3
4. √136

SOLUTION:

We can solve this problem with Heron’s formula but there is a simpler way

We know the length of the 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)

Therefore, Here D is the correct answer.

You should also try to solve using heron’s formula

For example, the Perimeter of an △ABC is 15. All sides have integral lengths. How many triangles can be made like that?

1. 7
2. 9
3. 8
4. 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.

a = 1, Possible sides of triangles 1, 7, 7

If a = 2, possible sides of triangles 2, 6, 7

assume a = 3, possible sides of triangles 3, 6, 6 and 3, 5, 7

a = 4, possible sides of triangles 4, 4, 7 and 4, 5, 6

a = 5, possible sides of triangles 5, 5, 5

So a total of 7 triangles are possible.

Therefore, the answer is 7.

Hence Option A is the correct answer.

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