Mathematics# Isosceles triangle

## What is the isosceles triangle?

## Characteristics of the isosceles triangle

## Properties

## Elements

## Height

## Area

## Perimeter

## How much do the angles of an isosceles triangle measure?

## Congruent sides

## Calculate the sides of an isosceles triangle

## How many equal angles it has

## Examples

To begin explaining the **isosceles triangle**, we must also remember the definition of triangle. We call a **triangle** a **polygon** that has three sides and is determined by three points that are not collinear called **vertices**. We must also remember that vertices are identified through **letters**, which are A, B and C. An isosceles triangle is a type of triangle that has at least two of its equal sides. This property is equivalent to **two angles** of the triangle that are equal. Therefore, an isosceles triangle has two equal sides and two equal angles. The name derives from the Greek **iso** (same) and **skelos** (leg).

**Related topics**

Catechus, hypotenuse, scalene triangle, right triangle, Pythagoras

The isosceles triangle is a type of **polygon** formed by **three sides**, this means that it is formed by three straight lines which will be **cut** two by two and in three points that are not **aligned**.

The main characteristics of the isosceles triangle are as follows:

- It is formed by three
**straight**lines; these straight lines will be cut two by two. - The points in which the straight lines are
**found**are known as vertices. - Each line
**segment**of the isosceles triangle is erected as the sides of the triangle. - The two continuous sides found in the isosceles triangle give rise to the
**inner angle**. - The triangle will be faced by three
**sides**as we said, by three**vertices**, by three**interior****angles**and by three**exterior angles**.

Like other triangles, the isosceles have their properties, which are:

The **angles opposite** the equal sides are equal. The **bisector** of the angle opposite the base cuts the base at its **midpoint**. The bisector always matches the median on the AB side. The bisector of the **opposite angle** to the base is **perpendicular** to the base. The bisector matches the corresponding height of the AB side.

As isosceles triangles have a pair of equal sides, this allows them also to have certain particularities quite recognized both with respect to **geometry** and in their **calculations**. Every equilateral triangle is also **isosceles**, but obviously this is not necessarily reciprocal.

In an isosceles triangle, there are also different elements that are part of it, among them we mention the following:

- Bisector
- Mediatrix
- Medium

The height (h) of the isosceles triangle can be calculated using the Pythagorean theorem. The sides a, b/2 and h form a right triangle. Sides b/2 and h are the legs and a hypotenuse.

**h ^{2} + (b/2)^{2} = a^{2} → h^{2} + ( b^{2}/4 ) = a^{2} → h^{2} = a^{2} – ( b^{2}/4 )**

Then getting another formula that tells us that the height of the isosceles triangle is:

**h = √( a ^{2} – ( b^{2}/4 ))**

The area that has an isosceles triangle can be calculated from the **base b** (the unrepeated side) and the **height** (h) of the **triangle** corresponding to the base. We can say then that, in this case, the area is the product of the **base** and the **height** divided by two, being its formula the following:

**Area = ( b · √( a ^{2} – ( b^{2}/4 )))/2**

The perimeter of an isosceles triangle can be obtained by **adding** the **three sides** of the **triangle**. Having two equal sides, the perimeter is twice the repeated side (a) plus the uneven side of the geometric figure.

**Perimeter = 2 · a + b**

Where a is one side repeated and b is the other side.

The isosceles right triangle has a **right angle** and two **acute angles** with a measure of **45°** each, thus, two sides of the triangle are equal and the other is different. The sides that are equal are known as the **cathetus** and the angle that is different is known as hypotenuse. It is symmetrical with respect to the height of the **hypotenuse**, which passes through the right angle.

**Congruence** is the moment when one of the sides of the triangle are congruent, so two angles can be congruent if they have the same measure and two strokes are congruent if they have the same **length**. In the triangle, there can be correspondence between the **sides** and between the **angles**. Remember that the isosceles triangle is the type of triangle that has two of its **equal** sides and a **different** one, having two sides with the **same measure**, is considered as congruent.

Isosceles triangles have two sides of equal measure and one of them has a different measure. It is also assumed that if two of the sides of an isosceles triangle are congruent or equal, this means that the angles opposite those sides will be congruent in the same way.

The isosceles triangle has two of its angles of equal measure, while the third will always be different.

An example of isosceles triangles would be as follows:

Written by Gabriela Briceño V.

Briceño V., Gabriela. (2019). *Isosceles triangle*. Recovered on 3 October, 2023, de Euston96: https://www.euston96.com/en/isosceles-triangle/