Triangles

A triangle is a polygon with three sides. It is one of the simplest shapes in geometry and has some interesting properties. The word "triangle" comes from Latin; "tri-" means three and "-angulus" means corner or angle. Therefore, a triangle is a shape that has three angles.

Basics of triangles

Before we discuss the types, properties and rules of triangles, let us understand the basic elements of a triangle.

• Sides: A triangle has three sides. These are the straight lines that form the boundaries of the triangle.
• Vertices: A triangle has three vertices. A vertex is the point where two sides of a triangle meet.
• Angles: A triangle has three angles. The sum of the interior angles in a triangle is always 180°.

Types of triangles

Triangles can be classified based on two criteria:

• In their favor.
• From their angles.

Classification based on sides

Equilateral triangle

An equilateral triangle is one in which the three sides are equal in length, and consequently, the three angles are also equal, each measuring 60°.

In the above figure, triangle ABC is equilateral with sides AB = BC = CA

Isosceles triangle

In an isosceles triangle two sides are of equal length and the angles opposite to these sides are also equal.

In triangle ABC, if AB = AC, then it is an isosceles triangle in which angles ∠ABC and ∠ACB are equal.

Scalene triangle

A scalene triangle is one in which the three sides are of different lengths. As a result, the three angles are also different.

In the above figure, triangle ABC does not have equal sides or angles.

Classification based on angles

Acute triangle

An acute-angled triangle is one whose all three interior angles are less than 90°.

Right triangle

A right triangle is one in which one angle is exactly 90°. The side opposite the right angle is the longest side and is called the hypotenuse.

In triangle ABC, the angle at C is 90°. Therefore, AB is the hypotenuse.

Obtuse-angled triangle

An obtuse-angled triangle is one in which one of the angles is more than 90°.

In this diagram, ∠ABC is greater than 90°, making ABC an obtuse triangle.

Properties of triangles

Angles of a triangle

As stated earlier, the sum of the interior angles of a triangle is always 180°. This fact is fundamental to understanding triangles and solving many problems involving them.

Suppose the angles of a triangle are A, B and C. Then the equation is:

A + B + C = 180°

Triangle inequality theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For a triangle with sides labeled a, b and c, the inequalities are:

a + b > c

a + c > b

b + c > a

Pythagorean theorem

The Pythagorean theorem applies to right triangles. It states that in a right triangle the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. If the hypotenuse is c:

c² = a² + b²

For example, in a right triangle:

a = 3, b = 4, c = 5

The Pythagorean theorem is as follows:

5² = 3² + 4²

so:

25 = 9 + 16

25 = 25

Congruence in triangles

Congruence means that two triangles have exactly the same size and shape. If two triangles are congruent, then their corresponding sides and angles are equal. There are several properties or criteria for triangle congruence.

Conformity criteria

• SSS (Side-Side-Side): If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.
• SAS (Side-Angle-Side): If two sides and the angle between them of a triangle are equal to two sides and the angle between them of another triangle, then the triangles are congruent.
• ASA (Angle-Side-Angle): If two angles and the included side of a triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.
• AAS (Angle-Angle-Side): If two angles and a nonconnected side of a triangle are equal to two angles and the corresponding nonconnected side of another triangle, then the triangles are congruent.
• RHS (Right Angle-Hypotenuse-Side): In right angled triangles, if the hypotenuse and a side of a triangle are equal to the hypotenuse and a side of the other triangle, then the triangles are congruent.

Area and perimeter of triangles

Circumference

The perimeter of a triangle is the sum of the lengths of its sides. If the sides of the triangle are a, b and c, then the perimeter P is calculated as:

P = a + b + c

Area

The general formula for the area of a triangle is:

Area = 0.5 × base × height

If the base is b and the height is h, then:

Area = 0.5 × b × h

For example, if the base of a triangle is 10 units and the height is 5 units then the area is:

Area = 0.5 × 10 × 5 = 25 square units

Heron's formula for area

If the sides of a triangle are known, we can use Heron's formula to find the area. According to Heron's formula:

First, calculate the semi-perimeter s of the triangle:

s = (a + b + c) / 2

Then, region A is given by:

A = √[s(s - a)(s - b)(s - c)]

Let us consider a triangle with sides 7, 8 and 9 units:

Semi-circumference:

s = (7 + 8 + 9) / 2 = 12

Using Heron's formula:

A = √[12(12-7)(12-8)(12-9)] = √[12×5×4×3] = √720 ≈ 26.83 square units

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, and they all meet at a point called the centroid. The centroid divides each median into two parts, one of which is twice the length of the other.

Summary

Triangles are fundamental figures in geometry, with unique properties and characteristics that make them interesting to study. They can be classified based on their sides and angles. From the sum of 180° angles to congruence criteria, the Pythagorean theorem in right triangles, and calculations of area and perimeter; triangles offer a wealth of fascinating mathematical challenges and applications.

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