A two-dimensional closed plane figure formed by joining three straight lines is referred to as a triangle. In Geometry, the concept of triangle is defined as “A closed figure or a polygon formed when three non-collinear points on a two-dimensional plane are joined.”
Vertices, Sides, and Angles are the basic elements of a triangle. In a triangle, there are 3 vertices, 3 sides, and 3 angles. Sides can also be termed as edges.
Are all triangles same? What difference do you observe in the triangles below?
Triangles are classified based on two criteria: Sides and Angles
Types of triangles based on sides
Triangles are classified into scalene triangle, equilateral triangle, and isosceles triangle depending on the number of identical sides.
Types of triangles based on angles
Triangles are classified into acute angled triangle, obtuse angled triangle, equiangular triangle, and right angled triangle depending on the measure of the angles.
Properties of Triangles
Every triangle has its own unique property. In general, triangles have certain properties that are common to any type of a triangle. These properties are:
1. The sum of the angles of a triangle is always equal to 180°.
2. The sum of the lengths of any two sides of a triangle is always greater than the third side.
3. A triangle can have only one right angle and the other two angles in the same triangle sum up to 90°.
4. Sides opposite to two equal angles will always be equal in lengths.
Altitude, Median, and Perpendicular bisector
Altitude: Every triangle can have three sides, any side can be considered as a base of a triangle. Thus, every triangle can have three bases. A perpendicular line segment from vertex to its opposite side (or extended line of the opposite side) is called altitude of a triangle. Every triangle will have three altitudes as shown in the image below.
Median: A line segment from vertex to midpoint of its opposite side is called a median of a triangle. Median bisects the side of a triangle.
Perpendicular bisector: A line perpendicular to a triangle’s side and passing through its midpoint is called the perpendicular bisector. Every triangle has three perpendicular bisectors.
Angular bisector: A straight line that divides a triangle’s angle into two equal angles is called the angle bisector.
Orthocenter: The point at which all three altitudes of a triangle meet is called orthocenter.
Orthocenters will not lie always inside the triangle. Observe the table below.
Centroid: The point at which all three medians of a triangle meet is called centroid. A centroid is considered to be a center of gravity as it is the center of the object. It lies inside the triangle.
In-center and Circumcenter: The point in the plane that is equidistant from its three vertices is called circumcenter.
The incenter of a triangle is the equidistant point on the interior of the triangle from all three sides.
The positions of orthocenter, centroid, circumcenter, and incenter in a triangle are listed in a table below.
Thales is credited with discovering five geometric theorems:
1. A circle is bisected by its diameter.
2. Angles in a triangle opposite two sides of equal length are equal.
3. Opposite angles formed by intersecting straight lines are equal.
4. The angle inscribed inside a semicircle is a right angle.
5. A triangle is determined if its base and two angles at the base are given.
While these may appear to be too simple to be revolutionary now, they actually provide a lot of information and were considered a huge breakthrough at the time. Thales’ theorems are used to compute specific longitude and proportional relationships in geometric figures with parallel lines. When two parallel lines are present, they are also used to calculate numerous trigonometric concepts. Thales derived these theorems while estimating the height of a pyramid, according to mythology. To do so, the mathematician computed the pyramid’s shadow on the ground. Thales used a cane to determine the measurements of the Egyptian pyramid in reference to the shadow cast by his cane.
Many countries do not give Thales credit for some of his theorems. One of his theorems, for example, is known in English as the Theorem of Interception, whereas in German it is known as the Theorem of Rays. However, none of them are exactly the same, and it resembles Pythagoras’ hypothesis better.
Know more about other concepts of Geometry on Geometric Shapes, Herons Formula, Area of Triangle, Perimeter of Circle and What is Perimeter.