Isosceles Triangles

Picture yourself in Paris, France in front of the Eiffel Tower and observe its structure.

Eiffel Tower and isosceles triangle, StudySmarter Originals

From above, we see that the structure of the Eiffel Tower represents a triangle. Now further inspect the dimensions of this triangle. Notice how the two opposite sides are equal while the base is different. This means we can rule out the Eiffel Tower being the shape of an equilateral triangle, which you recall is a triangle of three equal sides. So what kind of triangle might this be? To answer your question, this is called an isosceles triangle.

Components of an Isosceles Triangle

Consider the isosceles triangle ABC below.

Isosceles triangle, StudySmarter Originals

Here are the most important components of an isosceles triangle:

Sides and vertices

• The legs of the isosceles triangle are represented by the variable a.

• The base is defined by the variable b.

• The vertex $C$ is the topmost section of the isosceles triangle. This is also called the apex.

• The altitude$CD$is a perpendicular line segment drawn from the vertex to the base of an isosceles triangle.

• The length of the altitude is called height and is described by the variable h.

Angles

• The angle ${}_{\angle }C$ between the legs of the isosceles triangle is called the vertex angle (or apex angle).

• Each of the two angles ${}_{\angle }B$ and ${}_{\angle }A$ between a leg and the base of the isosceles triangle is called base angle.

Properties of an Isosceles Triangle

There are several significant properties of isosceles triangles you should familiarise yourself with in order to fully understand the composition of an isosceles triangle. The table below describes this in detail.

Identifying the Leg and Base of an Isosceles Triangle

Say we are given a triangle with three sides. We are told that the triangle is indeed an isosceles triangle. However, we need to determine which sides are the legs of the isosceles triangle and which side is the base.

To determine the legs of the isosceles triangle, take note of the characteristics below:

• Exactly two equal sides are present, which are the legs

• Both legs branch out from the vertex of the triangle

• The altitude is adjacent to the two legs

In contrast, the base should fulfil the following properties.

• The two angles at the ends of the base are equal

• An altitude drawn from the vertex is perpendicular to the base

• A perpendicular line segment through the vertex bisects the base, i.e. cuts the base into two equal halves

Isosceles Triangle Theorems

With that in mind, let us now discuss two notable theorems involving isosceles triangles that take a closer look into the two main properties as described above.

Theorem 1

The angles opposite the equal sides of an isosceles triangle are equal.

Theorem 2

The sides opposite the equal angles of an isosceles triangle are equal.

Types of Isosceles Triangles

There are three types of isosceles triangles to consider, namely

1. Isosceles acute;

2. Isosceles right;

3. Isosceles obtuse.

The table below compares each of these types of isosceles triangles.

Formulas of Isosceles Triangles

In this section, we shall look at three important formulas involving isosceles triangles, namely

1. The height of an isosceles triangle;

2. The perimeter of an isosceles triangle;

3. The area of an isosceles triangle.

The Height of an Isosceles Triangle

The height of an isosceles triangle can be found by applying Pythagoras Theorem. Say we have the isosceles triangle ABC below where the measures of a leg a and the base b are given.

Height of an isosceles triangle, StudySmarter Originals

We know that the altitude (line segment CD) from the vertex angle bisects the base of the isosceles triangle. This means that

$AD=BD=\frac{1}{2}b$.

Furthermore, ADC and BDC are right-angled triangles where a is the hypotenuse. Thus, to find the height, we can simply adopt Pythagoras Theorem as

id="2851376" role="math" $$\begin{gathered} a^2=h^2+{\left(\frac{b}{2}\right)}^2 \\ \Rightarrow h^2=a^2-{\left(\frac{b}{2}\right)}^2 \\ \Rightarrow h=\sqrt{a^2-{\left(\frac{b}{2}\right)}^2} \end{gathered}$$

The Perimeter of an Isosceles Triangle

The perimeter of an isosceles triangle is given by the following formula.

$P=2a+b$

where a is the length of the two equal sides and b is the base of the isosceles triangle. Let us demonstrate this with a worked example.

The Area of an Isosceles Triangle

Once you know the height of an isosceles triangle, calculating the area is a breeze. The formula for this is

$Area=\frac{1}{2}\times b\times h$,

where b is the base and h is the height of the isosceles triangle. Below is a worked example applying this method.

Altitudes in Isosceles Triangles

Let us now define an altitude of a triangle.

Now that we have established the definition of an altitude, we shall now link this idea with our subject at hand. The following are two theorems that relate the altitude to isosceles triangles.

Theorem 1

The altitude to the base of an isosceles triangle bisects the vertex angle.

Theorem 2

The altitude to the base of an isosceles triangle bisects the base.

Worked Examples Involving Isosceles Triangles

Comparing Triangles

There are three types of triangles we shall often see throughout this syllabus, namely

1. Isosceles triangle

2. Equilateral triangle

3. Scalene triangle

In this final section, we shall look at the differences between these three triangles. By familiarising ourselves with these contrasts, we can properly distinguish each type we are dealing with and perform the correct calculations. The table below compares these three triangles with respect to sides, angles and altitudes.