# Geometry

**Geometry**

In quantitative ability, geometry plays a significant role and involves solving problems related to various geometrical shapes and concepts. Some common topics in geometry that appear in quantitative ability tests or exams include:

**Basic Geometric Shapes:** Problems involving properties, perimeter, area, and relationships of basic shapes like triangles, rectangles, squares, circles, and quadrilaterals.

**Angles and Lines:** Questions on angles, parallel lines, perpendicular lines, and transversals Calculating missing angles or identifying angle relationships in geometric figures.

Example 1: Identifying Vertical Angles

If two intersecting lines form four angles, and one of the angles measures 45 degrees, what is the measure of the angle that is vertical to the 45-degree angle?

Solution:

Vertical angles are congruent, that is, they have equal measures. So, the angle that is vertical to the 45-degree angle will also measure 45 degrees.

Example 2: Complementary Angles

Two angles are complementary, and one of the angles measures 40 degrees. Find out the measurement of the other angle.

Solution:

Complementary angles add up to 90 degrees.

Let the other angle be x degrees.

40° + x° = 90°

x° = 90° - 40°

x° = 50°

Example 3: Supplementary Angles

Two angles are supplementary, and one of the angles measures 120 degrees. Find out the measurement of the other angle.

Solution:

Supplementary angles add up to 180 degrees.

Let the other angle be x degrees.

120° + x° = 180°

x° = 180° - 120°

x° = 60°

**Polygons:** polygons are two-dimensional shapes with straight sides. Here are some examples of polygons:

Triangle: A polygon with three sides. Examples include equilateral, isosceles, and scalene triangles.

Quadrilateral: A polygon with four sides. Examples include the square, rectangle, parallelogram, trapezoid, and rhombus.

Pentagon: A polygon with five sides.

Hexagon: A polygon with six sides.

Heptagon: A polygon with seven sides.

Octagon: A polygon with eight sides.

Nonagon: A polygon with nine sides.

Decagon: A polygon with ten sides.

Dodecagon: A polygon with twelve sides.

Example Question:

In a pentagon, two angles measure 90° each, and the remaining three angles measure 120° each. What is the total measure of all five angles in the pentagon?

Solution:

Let's assume that the two angles measuring 90° each are A and B, and the three angles measuring 120° each are C, D, and E.

Total measure of all five angles = A + B + C + D + E

Total measure of all five angles = 90° + 90° + 120° + 120° + 120°

The total measure of all five angles is 540°.

**Circles:** Questions about the properties of circles, including radius, diameter, circumference, and area. Tangents, chords, and arc measurements may also be covered.

Example Question:

A circular race track has a radius of 50 metres. If John runs exactly one lap around the track, how far did he run? (Use π ≈ 3.14)

Solution:

The distance covered by one lap around the circular track is equal to the circumference of the track.

Circumference = π * Diameter

Diameter = 2 * Radius

Diameter = 2 * 50 metres = 100 metres

Circumference = π * 100 metres ≈ 3.14 * 100 metres ≈ 314 metres

So, John ran approximately 314 metres.

**3D Geometry:** Problems involving three-dimensional shapes like cubes, cuboids, spheres, cylinders, cones, and pyramids Calculating volume, surface area, and other properties of these shapes.

Example Question 1:

Find the volume of a cube with edges measuring 5 centimetres.

Solution:

The formula for the volume of a cube is V = side^3, where "side" is the length of one edge.

V = 5 cm^3 = 5 cm * 5 cm * 5 cm = 125 cubic centimetres

So, the volume of the cube is 125 cubic centimetres.

Example Question 2:

Calculate the surface area of a rectangular prism with dimensions 6 cm × 4 cm × 3 cm.

Solution:

The formula for the surface area of a rectangular prism is SA = 2lw + 2lh + 2wh, where "l" is the length, "w" is the width, and "h" is the height.

SA = 2 (6 cm * 4 cm) + 2 (6 cm * 3 cm) + 2 (4 cm * 3 cm)

SA = 2 (24 cm^2) + 2 (18 cm^2) + 2 (12 cm^2)

SA = 48 cm^2 + 36 cm^2 + 24 cm^2

SA = 108 cm^2

So, the surface area of the rectangular prism is 108 square centimetres.

**Similarity and Congruence:** Questions related to similar and congruent figures Finding missing side lengths or angles using similarity or congruence theorems.

Example Question:

Triangle ABC and triangle DEF are similar triangles. The ratio of their perimeters is 3:5. If the perimeter of triangle DEF is 30 cm, find the perimeter of triangle ABC.

Solution:

Let the perimeter of triangle ABC be 3x, where x is a constant.

The ratio of the perimeters of triangle ABC and triangle DEF is 3:5, so:

Perimeter of triangle ABC/Perimeter of triangle DEF = 3/5

(3x) / 30 = 3/5

Cross-multiply:

5 * 3x = 3 * 30

15x = 90

x = 90 / 15

x = 6

So, the perimeter of triangle ABC is 3x = 3 * 6 = 18 cm.

**Coordinate Geometry:** Problems involving points, lines, and shapes in the coordinate plane Finding distances, slopes, or equations of lines.

Example Question 1:

Find the distance between points A (3, 4), and B (7, 8).

Solution:

The distance between two points with coordinates (x1, y1) and (x2, y2) can be found using the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

In this case:

x1 = 3, y1 = 4

x2 = 7, y2 = 8

Distance = √((7 - 3)^2 + (8 - 4)^2)

Distance = √(4^2 + 4^2)

Distance = √(16 + 16)

Distance = √32 ≈ 5.66

So, the distance between points A and B is approximately 5.66 units.

**Mensuration:** Problems that require calculating lengths, areas, volumes, and surface areas of geometric figures in practical scenarios.

Example Question:

Find the volume of a cube with edges measuring 3 centimetres.

Solution:

The volume of a cube is given by the formula: Volume = side^3

In this case, side = 3 cm.

Volume = 3 cm * 3 cm * 3 cm = 27 cubic cm

So, the volume of the cube is 27 cubic centimetres.

**Examples:**

**Example 1:** Perimeter of a Rectangle

Find the perimeter of a rectangle with a length of 12 cm and a width of 8 cm.

Solution:

The perimeter of a rectangle is 2 * (length + width).

Perimeter = 2 * (12 cm + 8 cm)

Perimeter = 2 * 20 cm

Perimeter = 40 cm

**Example 2:** Area of a Circle

Find out the area of a circle with a radius of 5 cm. (Use π = 3.14)

Solution:

Area of a circle = π * (radius)^2

Area = 3.14 * (5 cm)^2

Area = 3.14 * 25 cm^2

Area = 78.5 cm^2

**Example 3:** Volume of a Cylinder

Find the volume of a cylinder with a height of 10 cm and a radius of 4 cm. (Use π = 3.14)

Solution:

Volume of a cylinder = π * (radius)^2 * height

Volume = 3.14 * (4 cm)^2 * 10 cm

Volume = 3.14 * 16 cm^2 * 10 cm

Volume = 502.4 cm^3

**Example 4**: Diagonal of a Square

Find the length of the diagonal of a square with sides of length 6 cm.

Solution:

Using the Pythagorean theorem:

Diagonal^2 = Side^2 + Side^2

Diagonal^2 = 6 cm^2 + 6 cm^2

Diagonal^2 = 36 cm^2 + 36 cm^2

Diagonal^2 = 72 cm^2

Diagonal = √(72) cm

Diagonal ≈ 8.49 cm

These examples showcase different types of geometry problems commonly encountered in quantitative ability tests or exams. Solving these problems requires a good understanding of geometric principles, formulas, and the ability to apply them effectively. Regular practise will help you build confidence and improve your geometry skills and quantitative ability.

To excel in geometry with quantitative ability, it's essential to understand the properties and formulas associated with different shapes, angles, and measurements. Visualising and drawing figures can be helpful in solving geometry problems. Regular practise and familiarity with the different types of questions will improve your problem-solving skills and accuracy in geometry-related quantitative ability tasks.