ISOSCELES TRIANGLE: CHARACTERISTICS, FORMULA AND AREA, CALCULATION - MATHS - 2025

An 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 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 (they have the same length).

Based on the amplitude of the interior angles, isosceles triangles are classified as:

• Isosceles right triangle: two of its sides are equal. One corner is straight (90 ^or) and the others are the same (45 ^or each)
• Isosceles obtuse triangle: two of its sides are equal. One of the angles is obtuse (> 90 ^or).
• Isosceles acute triangle: two of its sides are equal. All angles are acute (<90 ^or) where both have the same measure.

Components

• The median: it 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 barycenter or centroid.
• The bisector: it 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 bisector: it is a segment perpendicular to the side of the triangle, which has its origin in the middle of it. There are three mediatices 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 side that is opposite and also this line is perpendicular to that side. All triangles have three heights, which coincide at a point called the orthocenter.

Properties

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 180 ^°.

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

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

"If a triangle has two congruent sides, the angles opposite those sides will also be congruent." Therefore, if a triangle is isosceles the angles of its bases 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:

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

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

Height, median, bisector, and bisector are coincident

The line that is drawn from the vertex opposite the base to the midpoint of the base of the isosceles triangle, is at the same time the height, the median and the bisector, as well as the bisector relative to the opposite angle of the base.

All these segments coincide in 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 the median AM is obtained, which is relative to vertex A and side BC.

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

Therefore, the bisector will always be equal to the median and vice versa.

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:

Med. (AMB) + Med. (AMC) = 180 ^or

2 _* Med. (AMC) = 180 ^or

Med. (AMC) = 180 ^or ÷ 2

Med. (AMC) = 90 ^or

It can be known that the angles formed by the AM segment 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 bisector, knowing that M is the midpoint.

Therefore the line AM:

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

Relative heights

Heights that are relative to equal sides have the same measurement as well.

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

Ortocenter, barycenter, incenter, and coincident circumcenter

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

How to calculate the perimeter?

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

As 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, it divides the triangle into two equal parts as it extends to the opposite vertex.

The height represents the opposite leg (a), the middle 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:

a ² + b ² = c ²

Where:

a ² = height (h).

b ² = b / 2.

c ² = side a.

Substituting these values ​​in the Pythagorean theorem, and solving the height, we have:

h ² + (b / 2) ² = a ²

h ² + b ² /4 = a ²

h ² = a ² - b ² /4

h = √ (a 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 ​​the triangles is always calculated with the same formula, multiplying the base by height and dividing by two:

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 you need to know at least the measure of the height or one of its angles.

Knowing the height, the Pythagorean theorem is used:

a ² + b ² = c ²

Where:

a ² = height (h).

c ² = side a.

b ² = b / 2, is unknown.

We isolate b ² from the formula and we have:

b ² = a ² - c ²

b = √ a ² - c ²

Since this value corresponds to half the base, it must be multiplied by two to obtain the complete measure of the base of the isosceles triangle:

b = 2 _* (√ a ² - c ²)

In the case 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.

In this 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 the base can be determined:

Exercises

First exercise

Find the area of ​​the isosceles triangle ABC, knowing that two of its sides are 10 cm and the third side is 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 of the isosceles triangle:

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

The values ​​are substituted in the formula:

Second exercise

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

Solution

In this case, the measurements 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 perpendicular to it is drawn, dividing the angle into two equal parts, one for each right triangle that is formed.

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

Now by trigonometry the value of half of the base is calculated, which corresponds to half of 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 already been determined.

By trigonometry it will be:

The perimeter is calculated:

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

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

P = 84 cm + 76 cm

P = 160 cm.

Third exercise

Calculate the internal angles of the isosceles triangle, knowing that the angle of the base is  = 55 ^or

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 every triangle will always be = 180 ^or:

 + Ê + Ô = 180 ^or

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

 = Ô

Ê = 55 ^or

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

55 ^or + 55 ^or + Ô = 180 ^or

110 ^or + Ô = 180 ^or

Ô = 180 ^o - 110 ^o

Ô = 70 ^o.

References

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3. Angel, AR (2007). Elementary Algebra. Pearson Education.
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6. José Jiménez, LJ (2006). Math 2.
7. Tuma, J. (1998). Engineering Mathematics Handbook. Wolfram MathWorld.