Triangles

Objective:

This topic gives an overview of

  • Triangles
  • Types of Triangles
    • Scalene triangle
    • Isosceles triangle
    • Equilateral triangle
    • Right triangle
    • Obtuse triangle
    • Acute triangle
  • Medians of a Triangle
  • Altitudes of a Triangle

Triangle

A triangle is a closed figure made of three line segments. Every triangle has three sides, three angles, and three vertices. These are known as the parts of a triangle. The sides and the angles of every triangle differ from one another; therefore, they do not look alike. 

Types of Triangles

Types of triangles may be classified by their sides, by their angles or by a combination of both sides and angles.

Triangles classified by their sides:

Scalene triangle:

A scalene triangle is a triangle that has no equal sides. The following is a scalene triangle.

Isosceles triangle:

An isosceles triangle is a triangle that has two equal sides. In an isosceles triangle, the angles opposite the equal sides are called the base angles, and they are equal. The following is an isosceles triangle.

 Equilateral triangle:

An equilateral triangle is a triangle that has three equal sides. All the three angles of an equilateral triangle are also equal, and each measures 60°. The following is an equilateral triangle.

 

Triangles classify by their angles:

Right triangle:

A triangle with one of its angles equal to 90° is known as a right-angled triangle. The side opposite the 90° angle is called the hypotenuse, and is the longest side of the triangle.he following is a right triangle.

 

Obtuse triangle:

An obtuse triangle has one angle that is bigger than 90 degrees (Obtuse angle). The following is an obtuse triangle.

Acute triangle:

In an acute triangle, all angle are less than 90 degrees, so all angles are acute angles.The following is an acute triangle.

We can also name triangles using angles and sides at the same time. If a triangle has one right angle and two equal sides, we can call that triangle right isosceles triangle. If a triangle has only acute angles and no equal sides, we can call that triangle acute scalene triangle. If a triangle has two equal sides and one obtuse angle, we can call that triangle obtuse isosceles triangle.

Notice that an angle cannot be obtuse and equilateral at the same time. An equilateral triangle cannot have an obtuse angle because all 3 angles in an equilateral triangle measure 60 degrees.

Medians of a Triangle

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. A triangle therefore has three medians.

A median of a triangle is a line segment from a vertex of the triangle to the midpoint of the side opposite that vertex. Because there are three vertices, there are of course three possible medians.The medians pass through a common point. Hence, the mediansof a triangle are concurrent. This point of concurrence is called the centroid, and is denoted by G. The centroid and medians of a triangle always lie inside the triangle. The centroid of a triangle divides the median in the ratio 2:1.

Altitudes of a Triangle

Make a triangular shaped cardboard ABC. Place it upright, on a table. The height is the distance from vertex A to the base BC. From A to BC you can think of many line segments.The height is given by the line segment that starts from A comes straight down to BC, and is perpendicular to BC. This line segment AL is an altitude of the triangle.

An altitude has one end point at a vertex of the triangle and the other on the line containing the opposite side. Through each vertex, an altitude can be drawn.

 

Summary:

  • The six elements of a triangle are its three angles and the three sides.
  • The line segment joining a vertex of a triangle to the mid point of its opposite side is called a median of the triangle. A triangle has 3 medians.
  • The perpendicular line segment from a vertex of a triangle to its opposite side is called an altitude of the triangle. A triangle has 3 altitudes.
  • A triangle is said to be equilateral if each one of its sides has the same length. 
  • In an equilateral triangle, each angle has measure 60°.
  • A triangle is said to be isosceles if atleast any two of its sides are of same length.
  • The non-equal side of an isosceles triangle is called its base; the base angles of an isosceles triangle have equal measure.

 

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