Properties of Triangles

Triangles are fundamental figures in the field of geometry that have important properties and applications. Understanding these properties is the key to solving complex geometric problems. Let's explore the properties of triangles, using illustrations and examples to clarify the concepts.

Definition of a triangle

A triangle is a three-sided polygon with three edges (or sides) and three vertices (or corners). The sum of the angles in a triangle is always 180°.

In the above diagram, triangle ABC shows vertices A, B, and C. The sides are the segments connecting these vertices: AB, BC, and CA.

Types of triangles according to sides

• Equilateral triangle: All three sides are the same length, and all angles are equal to 60°.

  An equilateral triangle has sides of equal length and each angle is 60°.

• Isosceles triangle: Two sides are of equal length, and the angles opposite to these sides are also equal.

  In an isosceles triangle two sides and two angles are equal.

• Scalene triangle: All sides and all angles are different.

  All the sides and angles of a scalene triangle have different measures.

Types of triangles based on angles

• Acute triangle: All angles are less than 90°.

  An acute-angled triangle has all angles less than 90°.

• Right angle: An angle exactly 90°.

  One angle of a right triangle is 90°.

• Obtuse triangle: One angle is greater than 90°.

  An obtuse triangle has one angle greater than 90°.

Triangle inequality theorem

The triangle inequality theorem states:

The sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side.

For a triangle ABC with sides a, b, and c:

a + b > c
b + c > a
a + c > b

Properties based on angle sum

The sum of all the interior angles of a triangle is always 180°. This is a fundamental property of all triangles. Let the angles be A, B, and C. Then:

A + B + C = 180°

Example: If two angles of a triangle are 45° and 85°, then the third angle can be found as follows:

180° - (45° + 85°) = 50°

Pythagorean theorem

The Pythagorean theorem applies to right triangles and deals with the lengths of their sides. It is stated as follows:

In a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).

The formula is:

c² = a² + b²

Consider a triangle with sides a = 3, b = 4, and hypotenuse c. Applying the theorem:

c² = 3² + 4² = 9 + 16 = 25

On calculating c we get:

c = √25 = 5

Medians of a triangle

The median of a triangle is a line segment that connects the vertex to the midpoint of the opposite side. Every triangle has three medians, which intersect at a point called the centroid. This point divides each median into segments with a ratio of 2:1.

In the figure above, the medians intersect at the centroid, which is considered the "center of mass" of the triangle.

Height of a triangle

The altitude of a triangle is a perpendicular segment from a vertex to a line containing the opposite side. Every triangle has three altitudes, which can be inside or outside the triangle, depending on the type of triangle.

The green line is the height from the top vertex of this triangle. It meets the opposite side at a right angle.

Orthocenter of a triangle

The orthocenter is the point where the three altitudes of a triangle intersect. It can be inside or outside the triangle.

Example: In a right triangle, the orthocenter lies at the vertex of the right angle.

In the above right triangle, the heights intersect at the right vertex.

Conclusion

Triangles are fundamental geometric shapes, full of properties and applications. By understanding their types, theorems such as the Pythagorean theorem, and different centers such as the centroid and orthocenter, students can solve complex problems and understand geometry more deeply.

Comments