# Chapter 5: Triangles and Vectors

**At Grade**Created by: CK-12

## Introduction

Finding unknown quantities in triangles, such as the lengths of sides and the measure of angles, is a critical portion of the study of trigonometry. This Chapter deals with how to find such unknown quantities in different cases where certain quantities are known.

Also introduced in this Chapter are vectors. Consider for a moment the idea that all things cannot be represented with only a number. For example, while you can count the number of pencils in your desk with a number, you cannnot completely describe the force you apply when pushing on your desk with just a number. This is because, in addition to the number describing the strength of the force, you need something describing the direction of the force (in this case, pushing down). Vectors are a way to describe these kinds of quantities; they have both a magnitude and a direction.

- 5.1.
## Sides of an Oblique Triangle

- 5.2.
## Determination of Unknown Angles Using Law of Cosines

- 5.3.
## Identify Accurate Drawings of Triangles

- 5.4.
## Derivation of the Triangle Area Formula

- 5.5.
## Heron's Formula

- 5.6.
## Determination of Unknown Triangle Measures Given Area

- 5.7.
## Angle-Angle-Side Triangles

- 5.8.
## Angle-Side-Angle Triangles

- 5.9.
## Possible Triangles with Side-Side-Angle

- 5.10.
## Law of Sines

- 5.11.
## Law of Cosines

- 5.12.
## General Solutions of Triangles

- 5.13.
## Directed Line Segments

- 5.14.
## Vector Addition

- 5.15.
## Vector Subtraction

- 5.16.
## Resultant of Two Displacements

- 5.17.
## Vector Multiplied by a Scalar

- 5.18.
## Translation of Vectors and Slope

- 5.19.
## Unit Vectors and Components

- 5.20.
## Resultant as the Sum of Two Components

- 5.21.
## Resultant as Magnitude and Direction

### Chapter Summary

## Summary

This Chapter introduced ways to find unknown quantities in triangles when other quantities are known. Included were derivations and applications of the Law of Sines and the Law of Cosines. This was followed by alternate formulas for finding the area of a triangle, including Heron's Formula.

Cases where quantities were known, such as two angles and the not included side, or two angles and the included side, were presented, along with methods to find the other unknown quantities. This led to the "ambiguous case", where triangles with two sides and the not included angle as known quantities were addressed.

The Chapter then turned to vectors, including their definition, translation, addition, subtraction, multiplication by a scalar, and decomposition into combinations of unit vectors.