**What is a scalene triangle?**

A **scalene triangle** is a polygon with three sides, where all have different measures or lengths; for that reason it was given the name scalene, which in Latin means unequal.

Triangles are polygons considered to be the simplest in geometry, because they are made up of three sides, three angles and three vertices. In the case of the scalene triangle, because it has all the different sides, it implies that its three angles will also be different.

**Characteristics of scalene triangles**

Scalene triangles are simple polygons because none of their sides or angles have the same measure, unlike isosceles and equilateral triangles.

Because all of their sides and angles have different measures, these triangles are considered irregular convex polygons.

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

**Scalene right triangle**: all its sides are different. One of its angles is right (90o) and the others are acute and with different measures.

**Scalene obtuse triangle**: all its sides are different and one of its angles is obtuse (> 90o).

**scalene acute triangle**: all its sides are different. All its angles are acute (< 90o), with different measures.

Another characteristic of scalene triangles is that due to the incongruity of their sides and angles, they do not have an axis of symmetry.

**Components/elements**

**median**

It is a straight 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**

It is a ray that divides each angle into two angles of equal measure. The bisectors of a triangle meet at a point called the incenter.

**the mediatrix**

It is a segment perpendicular to the side of the triangle, which has origin in the middle of it. There are three perpendicular bisectors in a triangle and they meet at a point called the circumcenter.

**The height**

It is the line that goes from the vertex to the opposite side, and also this line is perpendicular to said side. All triangles have three altitudes that coincide at a point called the orthocenter.

**Properties of the scalene triangle**

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

**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.

**incongruous sides**

All the sides of the scalene triangles have different measures or lengths; that is, they are incongruent.

**incongruent angles**

Since all the sides of the scalene triangle are different, its angles will also be different. However, the sum of the internal angles will always be equal to 180º, and in some cases, one of its angles may be obtuse or right, while in others all of its angles will be acute.

**height, median, bisector and bisector are not coincident**

Like any triangle, the scalene has various line segments that compose it, such as: height, median, perpendicular bisector and bisector.

Due to the particularity of its sides, in this type of triangle none of these lines will coincide in just one.

**Orthocenter, centroid, incenter, and circumcenter are not coincident**

As the height, median, bisector and perpendicular bisector are represented by different line segments, in a scalene triangle the meeting points –the orthocenter, barycenter, incenter and circumcenter– will be at different points (that is, they do not coincide).

Depending on whether the triangle is acute, right, or obtuse, the orthocenter has different locations:

to. If the triangle is acute, the orthocenter will be inside the triangle.

b. If the triangle is right-angled, the orthocenter will coincide with the vertex of the straight side.

c. If the triangle is obtuse, the orthocenter will be on the outside of the triangle.

**relative heights**

The heights are relative to the sides.

In the case of the scalene triangle, these heights will have different measures. Every triangle has three relative heights and to calculate them Heron’s formula is used.

**Calculation of perimeter, area, height and sides**

**How to calculate the perimeter?**

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

Since in this case the scalene triangle has all its sides with different measures, its perimeter will be:

P = side a + side b + side c.

**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 two:

Area = (base * h) ÷2

In some cases the height of the scalene triangle is not known, but there is a formula that was proposed by the mathematician Heron, to calculate the area knowing the measure of the three sides of a triangle.

Where:

a, b and c represent the sides of the triangle.

sp corresponds to the semiperimeter of the triangle, that is, half the perimeter:

sp = (a + b + c) ÷ 2

In the event that you only have the measure of two of the sides of the triangle and the angle that is formed between them, the area can be calculated by applying the trigonometric ratios. So you have to:

Area = (side * h) ÷2

Where the height (h) is the product of one side by the sine of the opposite angle. For example, for each side, the area will be:

Area = (b * c * sin A) ÷ 2

Area = (a * c * sin B) ÷ 2.

Area = (a * b * sin C) ÷ 2

**How to calculate the height?**

Since all the sides of the scalene triangle are different, it is not possible to calculate the height with the Pythagorean theorem.

From Heron’s formula, which is based on the measurements of the three sides of a triangle, the area can be calculated.

The height can be cleared from the general area formula:

The side is replaced by the measure of the side a, b or c.

Another way to calculate the height when the value of one of the angles is known is by applying the trigonometric ratios, where the height will represent one leg of the triangle.

For example, when the angle opposite the height is known, it will be determined by the sine:

**How to calculate the sides?**

When you have the measure of two sides and the angle opposite them, it is possible to determine the third side by applying the theorem of cosines.

For example, in a triangle AB, the altitude relative to segment AC is plotted. In this way the triangle is divided into two right triangles.

To calculate the side c (segment AB), the Pythagorean theorem is applied for each triangle:

For the blue triangle we have:

c2 = h2 + m2

Since m = b – n, substitute:

c2 = h2 + b2 (b – n)2

c2 = h2 + b2 – 2bn + n2.

For the pink triangle we have:

h2 = a2 – n2

It is substituted in the previous equation:

c2 = a2 – n2 + b2 – 2bn + n2

c2 = a2 + b2 – 2bn.

Knowing that n = a * cos C, it is substituted in the previous equation and the value of side c is obtained:

c2 = a2 + b2 – 2b* a * cos C.

By the Law of cosines, the sides can be calculated as:

a2 = b2 + c2 – 2b* c * cos A.

b2 = a2 + c2 – 2a* c * cos B.

c2 = a2 + b2 – 2b* a * cos C.

There are cases where the measurements of the sides of the triangle are not known, but rather its height and the angles formed at the vertices. To determine the area in these cases it is necessary to apply the trigonometric ratios.

Knowing the angle of one of its vertices, the legs are identified and the corresponding trigonometric ratio is used:

For example, leg AB will be opposite for angle C, but adjacent to angle A. Depending on the side or leg corresponding to the height, the other side is cleared to obtain its value.

**solved exercises**

**First exercise**

Calculate the area and height of the scalene triangle ABC, knowing that its sides are:

w = 8cm.

b = 12cm.

c = 16 cm.

**Solution**

As data, the measurements of the three sides of the scalene triangle are given.

Since the value of the height is not available, the area can be determined by applying Heron’s formula.

First the semiperimeter is calculated:

sp = (a + b + c) ÷ 2

sp = ( 8cm + 12cm + 16cm) ÷ 2

sp = 36 cm ÷ 2

sp = 18cm.

Now substitute the values in Heron’s formula:

Knowing the area, the height relative to side b can be calculated. From the general formula, solving it we have:

Area = (side * h) ÷ 2

46, 47 cm2 = (12 cm * h) ÷2

h = (2 * 46.47 cm2) ÷ 12 cm

h = 92.94 cm2 ÷ 12 cm

h = 7.75 cm.

**second exercise**

Given the scalene triangle ABC, whose measurements are:

Segment AB = 25 m.

BC segment = 15 m.

At vertex B an angle of 50º is formed. Calculate the height relative to side c, perimeter and area of that triangle.

**Solution**

In this case, you have the measurements of two sides. To determine the height it is necessary to calculate the measure of the third side.

Since the angle opposite to the given sides is given, it is possible to apply the law of cosines to determine the measure of side AC (b):

b2 = a2 + c2 – 2a*c * cos B

Where:

a = BC = 15 m.

c = AB = 25m.

b = AC.

B = 50o.

The data is replaced:

b2 = (15)2 + (25)2 – 2*(15)*(25) * cos 50

b2 = (225) + (625) – (750) * 0.6427

b2 = (225) + (625) – (482,025)

b2 = 367,985

b = √367,985

b = 19.18 m.

Since we already have the value of the three sides, the perimeter of that triangle is calculated:

P = side a + side b + side c

P = 15m + 25m + 19.18m

P = 59.18 m

Now it is possible to determine the area by applying Heron’s formula, but first the semi-perimeter must be calculated:

sp = P ÷ 2

sp = 59.18 m ÷ 2

sp = 29.59m.

Substitute the measures of the sides and the semiperimeter in Heron’s formula:

Finally, knowing the area, the height relative to side c can be calculated. From the general formula, clearing it, we have:

Area = (side * h) ÷ 2

143.63 m2 = (25 m * h) ÷ 2

h = (2 * 143.63 m2) ÷ 25 m

h = 287.3 m2 ÷ 25 m

h = 11.5 m.

**third exercise**

In the scalene triangle ABC, side b measures 40 cm, side c measures 22 cm, and vertex A forms an angle of 90o. Calculate the area of that triangle.

**Solution**

In this case, the measures of two sides of the scalene triangle ABC are given, as well as the angle formed at the vertex A.

To determine the area it is not necessary to calculate the measure of side a, since through the trigonometric ratios the angle is used to find it.

Since the angle opposite the height is known, it will be determined by the product of a side and the sine of the angle.

Substituting into the area formula, we have:

Area = (side * h) ÷2

h = c * sin A

Area = (b * c * sin A) ÷ 2

Area = (40 cm * 22 cm * sin 90) ÷ 2

Area = (40 cm * 22 cm * 1) ÷ 2

Area = 880 cm2 ÷ 2

Area = 440 cm2.

**References**

Alvaro Rendón, AR (2004). Technical drawing: activity book.

Angel Ruiz, HB (2006). Geometries. CR Technological, .

Angel, AR (2007). Elementary Algebra. Pearson Education,.

Baldor, A. (1941). Algebra. Havana: Culture.

Barbosa, JL (2006). Plane Euclidean Geometry. Rio de Janeiro,.

Coxeter, H. (1971). Fundamentals of Geometry. Mexico: Limusa-Wiley.

Daniel C. Alexander, G.M. (2014). Elementary Geometry for College Students. Cengage Learning.

Harpe, P.S. (2000). Topics in Geometric Group Theory….