Geometry

Geometry examines the sizes, shapes and distances of objects and compares the relationships between points, lines, curves, angles, surfaces and solids.

This topic is divided into two segments: Plane Geometry and Solid Geometry.

1. Plane Geometry: The study of flat surfaces in two-dimensional space
   • Example: lines, circles, polygons
2. Solid Geometry: The study of objects in three-dimensional space
   • Example: cubes, cylinders, spheres

In this topic, we shall observe the contents that comprise Plane Geometry and Solid Geometry to give us a better understanding of Geometry. For simplicity, these two segments will be further divided into smaller sections.

Plane Geometry

The idea of Plane Geometry is derived from looking at objects from a two-dimensional perspective. Let us take a look!

These concepts constitute the basic objects from which all Geometry can be constructed, in other words, any other geometrical object can be defined in terms of a combination of these three concepts.

Cartesian Coordinate System

Imagine we would like to study the properties of the triangle below. We would like to know the measure of its angles and sides as well as how much space it occupies (this is also known as the area). However, we were not given any suitable measuring instrument such as a ruler to determine this. How do we go about this problem then? Here is where the Cartesian Coordinate System enters the scene.

Example 1, Aishah Amri - StudySmarter Originals

A figure illustrated in a Cartesian coordinate system in two dimensions represented by the x-axis and y-axis. A point is denoted by the coordinates (x, y) on the Cartesian plane.

The two dimensions here refer to the length and height of the figure. The point of intersection between the x-axis and the y-axis is called the origin and is denoted by the letter O. The coordinates of the origin is (0, 0).

The cartesian coordinate system contains four quadrants, listed below.

1. Quadrant I: Refers to a point located in both positive regions of the x-axis and y-axis.

2. Quadrant II: Refers to a point located in the negative region of the x-axis and the positive region of the y-axis.

3. Quadrant III: Refers to a point located in both negative regions of the x-axis and y-axis.

4. Quadrant IV: Refers to a point located in the positive region of the x-axis and the negative region of the y-axis.

Below is a graphical representation of the Cartesian coordinate system.

Quadrant system, Aishah Amri - StudySmarter Originals

Let us return to our triangle, introduced at the beginning of this section. With these concepts in place, let us position this triangle on the Cartesian plane.

Example 2, Aishah Amri - StudySmarter Originals

Here, our triangle is represented by 3 points A, B and C and 3 line segments AB, AC and BC. With this information, we can definitely calculate the required measures for this triangle. Though more on this later For now, let us stick to our Quadrant system with the following example.

Lines

A line is represented by the equation y = mx + c, where m is the slope or gradient of the line and c is the y-intercept.

The gradient measures the steepness of a line and is given by the formula:

.

Two lines are said to be parallel if they lie on the same plane and do not intersect each other. A pair of parallel lines have the same slope.

Parallel lines, Aishah Amri - StudySmarter Originals

Two lines are said to be perpendicular if they intersect each other at right angles. The product of the two slopes is –1.

Perpendicular lines, Aishah Amri - StudySmarter Originals

A line segment is a line with two endpoints.

Line segment, Aishah Amri - StudySmarter Originals

A ray is a line with a fixed starting point an endpoint that goes on forever.

Ray, Aishah Amri - StudySmarter Originals

Midpoint and Distance Formula

Midpoint Formula locates the point that is equidistant from two endpoints on a line segment.

The Distance Formula calculates the length between two points on a line.

Angles

Angles are useful when it comes to describing polygons such as quadrilaterals and triangles as we shall see later on in this lesson. Let us first define an angle.

Below are several notable types of angles you should familiarize yourself with.

By the definition above, a vector quantity has both direction and magnitude (size). A vector can be visualised geometrically as a directed line segment with a length equal to the magnitude of the vector and a direction indicated by an arrow. Below is a graphical representation of a vector.

Vector, Aishah Amri - StudySmarter Originals

Let us now look at some common vector operations in the table below.

Perimeter and Area

Congruence and Similarity

Congruence and similarity serve an important role in Geometry when it comes to comparing shapes and finding the measures between them.

Transformations

In this section, we shall become acquainted with the concept of transformations. Transformations help us visualize objects on a plane in different orientations.

Symmetry

Symmetry is an important concept when it comes to reproducing shapes without changing their original form. Let us dive into its definition and become familiar with three primary types of symmetry as described in the table below.

Property	Description
Exterior Angle of a Polygon	The sum of the exterior angles of a polygon is 360o For a polygon with n sides, each exterior angle is equal to Exterior Angle =
Interior Angle of a Polygon	For a polygon with n sides, each interior angle of a polygon is given by the formula Interior Angle = 180o - Exterior Angle

Triangles

Triangles, as you shall see throughout Geometry, play an important role in another subtopic called Trigonometry. Though, more on that later! Here, we shall only cover the area of a basic triangle and describe the six main types of triangles we shall commonly see throughout this syllabus.

The area of a triangle is given by the formula

,

where b is the base and h is the height.

Area of a triangle, Aishah Amri - StudySmarter Originals

Quadrilaterals

Next, we shall look at another form of polygons called quadrilaterals. The table below describes several types of quadrilaterals along with their properties and area formula.

Type of Quadrilateral	Diagram	Properties	Area
Rectangle	Rectangle, Aishah Amri - StudySmarter Originals	Opposite sides are equal 4 right angles Opposite sides are parallel	lh
Square	Square, Aishah Amri - StudySmarter Originals	4 equal sides 4 right angles Opposite sides are parallel	l2
Trapezoid	Trapezoid, Aishah Amri - StudySmarter Originals	2 parallel sides
Parallelogram	Paralellogram, Aishah Amri - StudySmarter Originals	Opposite sides are equal Opposite sides are parallel	bh
Rhombus	Rhombus, Aishah Amri - StudySmarter Originals	4 right angles Opposite sides are parallel

Circles

Let us move on to another shape of interest called circles. Here, we shall also discover the components that make up a circle.

Solid Geometry

Let us now move on to the next vital section of this topic called Solid Geometry. Here, we shall visualize objects in three-dimensional space.

There are two types of solids to consider in this section.

1. Polyhedra: Any solid with only flat faces
   • Example: Cubes, Pyramids, Prisms
2. Non-Polyhedra: Any solid with at least one curved face
   • Example: Spheres, Cylinders, Cones

A solid is often illustrated in a Cartesian coordinate system in three dimensions represented by the x-axis, y-axis and z-axis. Below is a graphical representation of a sphere centred at the origin with a radius of 2 units. The red line represents the x-axis, the green line denotes the y-axis and the blue line defines the z-axis.

Three-dimensional Cartesian coordinate system of a sphere, Aishah Amri - StudySmarter Originals

Properties of Solids

All solids have two characteristics that define their form.

1. Surface area

2. Volume

Another way to distinguish different solids from each other is by observing the number of vertices, edges and faces they own.

Faces, Edges and Vertices

We shall first describe what these components mean for a solid and present a table illustrating several solids along with their number of faces, edges and vertices.

Surface Area and Volume

In this section, we shall exhibit a table that describes the formula of the surface area and volume of a few notable solids.

Cross-Section of a Solid

Another important concept that falls under the category of Solid Geometry is called the cross-section.

Euler's Formula

Euler's Formula states that for any polyhedron that does not intersect itself or have any holes, the number of faces plus the number of vertices minus the number of edges always equals two. This can be written by the expression below.

Let us look at an example that applies this formula.