**What is an isosceles triangle?**

A **isosceles triangle** is a polygon with three sides, where two of them have the same measure and the third side a different measure. This last side is called the base. Due to this characteristic it was given this name, which in Greek means «equal legs».

Triangles are polygons considered to be the simplest in geometry, because they are made up of three sides, three angles, and three vertices. They are the ones that have the least number of sides and angles with respect to the other polygons, however their use is very extensive.

**Characteristics of isosceles triangles**

The isosceles triangle was classified using the measure of its sides as a parameter, since two of its sides are congruent, that is, they have the same length.

According to the amplitude of the internal angles, isosceles triangles are classified as:

**Isosceles right triangle**: two of its sides are equal. One of its angles is right (90o) and the others are equal (45o each)

**Isosceles obtuse triangle**: two of its sides are equal. One of its angles is obtuse (> 90o).

**Isosceles acute triangle**: two of its sides are equal. All of its angles are acute (< 90o), where two have the same measure.

**Components**

**median**: is a line that starts from the midpoint of one side and reaches the opposite vertex. The three medians meet at a point called the centroid or centroid.

**the bisector**: is a ray that divides the angle of each vertex into two angles of equal measure. That is why it is known as the axis of symmetry, and this type of triangle has only one.

**the mediatrix**: is a segment perpendicular to the side of the triangle, which has its origin in the middle of it. There are three medians in a triangle and they meet at a point called the circumcenter.

**The height**: is the line that goes from the vertex to the side that is opposite and also this line is perpendicular to that side. All triangles have three altitudes, which coincide at a point called the orthocenter.

**Properties of isosceles triangles**

Isosceles triangles are defined or identified because they have several properties that represent them, originating from the theorems proposed by great mathematicians:

**internal angles**

The sum of the interior angles is always equal to 180o.

**sum of the sides**

The sum of the measures of two sides must always be greater than the measure of the third side, a + b > c.

**congruent sides**

Isosceles triangles have two sides with the same measure or length; that is, they are congruent, and the third side is different from these.

**congruent angles**

Isosceles triangles are also known as isoangular triangles, because they have two angles that have the same measure (congruent). These are located at the base of the triangle, opposite the sides that are the same length.

Due to this, the theorem was generated that establishes that:

«If a triangle has two congruent sides, the angles opposite those sides will also be congruent.» Therefore, if a triangle is isosceles, its base angles are congruent.

**Example:**

The following figure shows a triangle ABC. By drawing its bisector from the vertex of angle B to the base, the triangle is divided into two equal triangles BDA and BDC:

Thus, the vertex angle B was also divided into two equal angles. The bisector is now the common side (BD) between those two new triangles, while sides AB and BC are the congruent sides. Thus we have the case of congruence side, angle, side (LAL).

With this it is shown that the angles of the vertices A and C have the same measure, as well as it can be shown that since the triangles BDA and BDC are congruent, the sides AD and DC are also congruent.

**height, median, perpendicular bisector, and bisector are coincident**

The line drawn from the vertex opposite the base to the midpoint of the base of the isosceles triangle is both the altitude, the median, and the bisector, as well as the bisector relative to the opposite angle of the base.

All these segments coincide in only one that represents them.

**Example:**

The following figure shows the triangle ABC with a midpoint M that divides the base into two segments BM and CM.

By drawing a segment from point M to the opposite vertex, by definition we obtain median AM, which is relative to vertex A and side BC.

Since the segment AM divides the triangle ABC into two equal triangles AMB and AMC, it means that there will be the case of congruence side, angle, side, and therefore AM will also be the bisector of BÂC.

That is why the bisector will always be equal to the median and vice versa.

The segment AM forms angles that have the same measure for triangles AMB and AMC; that is, they are supplementary, in such a way that the measure of each one will be:

Md (AMB) + Md (AMC) = 180o

2 * Med. (AMC) = 180o

Md. (AMC) =180o ÷ 2

Md (AMC) = 90o

It can be known that the angles formed by the segment AM with respect to the base of the triangle are right, which indicates that this segment is totally perpendicular to the base.

Therefore it represents the height and the perpendicular bisector, knowing that M is the midpoint.

Therefore, the line AM:

It represents the height of BC. Is medium size. It is contained within the bisector of BC. It is the angle bisector of the vertex Â

**relative heights**

The heights that are relative to the equal sides have the same measure as well.

Since the isosceles triangle has two equal sides, their two respective altitudes will also be equal.

**Coincident orthocenter, centroid, incenter, and circumcenter**

Since the height, median, bisector and perpendicular bisector relative to the base are represented at the same time by the same segment, the orthocenter, centroid, incenter and circumcenter will be collinear points, that is, they will lie on the same line:

**Calculation of isosceles triangles**

**How to calculate the perimeter?**

The perimeter of a polygon is calculated by adding the sides.

Since in this case the isosceles triangle has two sides with the same measure, its perimeter is calculated with the following formula:

P = 2*(side a) + (side b).

**How to calculate the height?**

The height is the line perpendicular to the base, divides the triangle into two equal parts by extending to the opposite vertex.

The height represents the opposite leg (a), half of the base (b/2) the adjacent leg and the side “a” represents the hypotenuse.

Using the Pythagorean theorem, the value of the height can be determined:

*a2* + *b*2 = *c*2

Where:

*to*2 = height (h).

*b*2 = b / 2.

*c*2 = side a.

Substituting those values in the Pythagorean theorem, and clearing the height we have:

*h*2 + (*b* / 2)2 = *to*2

*h*2 + *b*2 / 4 = *to*2

*h*2 = *to*2 – *b*2 / 4

h = √ (*to*2 – *b*2 / 4).

If the angle formed by the congruent sides is known, the height can be calculated with the following formula:

**How to calculate the area?**

The area of triangles is always calculated with the same formula, multiplying the base times the height and dividing by 2:

There are cases where only the measurements of two sides of the triangle and the angle formed between them are known. In this case, to determine the area it is necessary to apply the trigonometric ratios:

**How to calculate the base of the triangle?**

Since the isosceles triangle has two equal sides, to determine the value of its base it is necessary to know at least the measure of the height or one of its angles.

Knowing the height, the Pythagorean theorem is used:

a2 + b2 = c2

Where:

a2 = height (h).

c2 = side a.

b2 = b / 2, is unknown.

We clear b2 from the formula and we have:

b2 = a2 – c2

b = √ a2 – c2

Since that value corresponds to half of the base, it must be multiplied by 2 to obtain the full measure of the base of the isosceles triangle:

b = 2 * (√ a2 – c2)

In the event that only the value of its equal sides and the angle between them are known, trigonometry is applied, drawing a line from the vertex to the base that divides the isosceles triangle into two right triangles.

That way half of the base is calculated with:

It is also possible that only the value of the height and angle of the vertex that is opposite the base are known. In that case, by trigonometry you can determine the base:

**Exercises **

**First exercise**

Find the area of the isosceles triangle ABC, knowing that two of its sides measure 10 cm and the third side measures 12 cm.

**Solution**

To find the area of the triangle, it is necessary to calculate the height using the area formula that is related to the Pythagorean theorem, since the value of the angle formed between the equal sides is not known.

We have the following data for the isosceles triangle:

Equal sides (a) = 10 cm. Base (b) = 12 cm.

Substitute the values into the formula:

**second exercise**

The length of the two equal sides of an isosceles triangle is 42 cm, the union of these sides forms an angle of 130o. Determine the value of the third side, the area of that triangle, and the perimeter.

**Solution**

In this case, the measures of the sides and the angle between them are known.

To know the value of the missing side, that is, the base of that triangle, a line is drawn perpendicular to it, dividing the angle into two equal parts, one for each right triangle that is formed.

Equal sides (a) = 42 cm. Angle (Ɵ) = 130o

Now, by trigonometry, the value of half the base is calculated, which corresponds to half the hypotenuse:

To calculate the area it is necessary to know the height of that triangle, which can be calculated by trigonometry or by the Pythagorean theorem, now that the value of the base has been determined.

By trigonometry it will be:

The perimeter is calculated:

P = 2*(side a) + (side b).

P = 2* (42cm) + (76cm)

D = 84cm + 76cm

D = 160 cm.

**third exercise**

Calculate the internal angles of the isosceles triangle, knowing that the base angle is Â= 55o

**Solution**

To find the two missing angles (Ê and Ô) it is necessary to remember two properties of triangles:

The sum of the internal angles of any triangle will always be = 180o:

Â + Ê + Ô = 180 or

In an isosceles triangle, the base angles are always congruent, that is, they have the same measure, therefore:

Â = Ô

Ê = 55o

To determine the value of angle Ê, substitute the values of the other angles in the first rule and solve for Ê:

55o + 55o + Ô= 180o

110 or + Ô = 180 or

Ô = 180 o – 110 o

Ô = 70o.

**References**

Alvarez, E. (2003). Elements of geometry: with numerous exercises and geometry of the compass. University of Medellin. Alvaro Rendón, AR (2004). Technical drawing: activity book. Angel, AR (2007). Elementary Algebra. Pearson Education. Arthur Goodman, LH (1996). Algebra and trigonometry with analytical geometry. Pearson Education. Baldor, A. (1941). Algebra. Havana: Culture. Jose Jimenez, LJ (2006). Mathematics 2. Tuma, J. (1998). Engineering Mathematics Handbook. Wolfram Math World.