50 Solved Mathematical Problems on Triangles

Introduction

If you have to draw three straight lines and put the endpoints of each together to form a shape, then you are making a so-called triangle. Triangle derived from the word "tri" means three and angle, together means three angles. Triangle is a three-angled, three-sided shape that regardless of the length of each side, forms an enclosed figure. This means that if the lines are not meeting together in their endpoints, cannot make an enclosed figure, hence, cannot be a triangle.

Triangles are one of the interesting subjects in trigonometry. May it be plane or spherical triangles, both serve to tickle the creativity and mathematical wit of anyone who tries to decipher it. We first recognized the shape of triangles when we are a toddler. When our mom had led us to play with blocks and solid figures to prepare us before entering the grade school. Now, we are already a youngster and already understood the essence of that three-angled figure.

We must understand the more complicated triangle problems before entering college, especially in the course of engineering. This subject is essential as a pre-requisite to major subjects in higher years. Having this in mind is a great contributory factor to battle the problems of Solid Geometry, Calculus, Advance Trigonometry, Physics and the likes. A strong and relevant foundation about the theories and formulas of triangular shapes will mitigate the sufferings of the dim years in college. (Just exaggerating!). Anyway, if you want to lessen the burdens of mathematics of trigonometry and other related matters, this book clearly suits for you.

Be calm, prepare your table and review materials, read this book deeply before solving Math problems. May this book serve a great help for you, youngsters of this modern digital generation.

Chapter 1

The Parts of Triangles

Before heading to a mathematical battle, knowing the terms and parts of triangles is important. We have plenty of this being described below and in the following pages.

Figure 1: The Triangle

This is a triangle, as you can see, there are several notations represented by letters for each side and vertices of the figure. As described, there are three lines connected by its endpoints together and forming a closed shape. The lines are the sides of the triangles hence, the segment AC, CB and BA. According to the diagram, the segment AC can also be named as the side b, segment CB as the side a and the segment BA, side c. The side a is the longest among the two followed by the c and then b, proving the fact that a triangle can be formed regardless of the lengths of its sides.

The vertices are the edgy parts of the triangle, whereas represented by the capital letters A, B and C. The vertices have interior angles in which if we add up the interior angles of the three vertices would sum up to 180°. Any triangle regardless of sizes and shapes have the sum of 180° of interior angles. 

Right triangle - Wikipedia Generally, there are just two types of triangles, the right triangle and the oblique triangle. Right triangles are those having a 90° interior angle while oblique triangle doesn’t have. Having a 90° angle means that the two adjacent sides are perpendicular to each other as you can see in the diagram below.

Figure 2: Right Triangle

In the above figure represented a triangle having the right interior angle. As stated, the side b and side a are perpendicular to each other hence, having an interior angle of 90°. In the case of right triangle, the side opposite the 90° angle is always the longest, known as the hypotenuse of the triangle. To know the length of each side, we are going to tackle the most famous equations in triangles, the Pythagorean formula. In this equation, the longest side or hypotenuse can be found from the square root of the sum of the squares of the two sides, in mathematical equation; c² = a² + b², where a, b and c are the sides of a right triangle.

Please refer to our first problems at the following pages, the easiest ones. The correct answer and its explanations can be found at the later parts of this chapter.

Another important part of the triangle is its interior angles. An angle is the separation of the two straight lines which shares the common endpoint. In triangle, interior angles represent the angles inside the triangle.

Figure 3: An Isosceles Triangle

Look at the on the recent page, angle B and C is 68°, if all of the interior angles of a triangle are summed up to 180°, what would be the angle A? Let’s take a little computation.

A + B + C = 180°

A + 68° + 68° = 180°

A + 136° = 180°

A = 180° - 136°

A = 44°

Mathematically speaking, the angle A must be 44°. We can now find a third angle if two interior angles of the triangle are given in the problem. But to clear things up, if the angle is the separation of the two lines sharing the common endpoint, what angle did they make if the lines are directly opposite to each other, or in layman’s term, what angle did a straight line have? Let’s look at the diagram below.

Figure 4: A Straight Line’s Angle

Imagine drawing a perpendicular line in the middle of that straight line, what can you see? Did you saw two perpendicular angles? As a matter of fact, a line’s angle is the combination of two perpendicular angles, hence, if summed up, equal to 180°.

Going back to triangle, there are just three types of triangles according to its interior angles, namely, the equilateral, isosceles