Triangles
are fascinating shapes found everywhere, from nature to architecture. But what exactly is a triangle in geometry? A triangle is a three-sided polygon with three edges and three vertices.
One of its key properties is that the sum of its three interior angles is always 180 degrees, known as the
angle sum property
.Triangles come in different types, such as equilateral, isosceles, and scalene, depending on the lengths of their sides.
They are often labeled as ∆ABC, where A, B, and C are the vertices. Read on to learn more about triangles!
Isosceles Triangle
What is a
Triangle
?
A triangle is a three-sided polygon, where each side is a straight line segment. The points where two sides meet are called vertices; each vertex forms an angle. Triangles are classified by their angles (acute, right, obtuse) and sides (equilateral, isosceles, scalene).
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Parts of a Triangle
A triangle consists of three sides, three angles, and three vertices. In triangle XYZ, the sides are XY, YZ, and ZX. The angles are formed where two sides meet, denoted by the symbol ∠.
The three angles in triangle XYZ are ∠XYZ, ∠YZX, and ∠ZXY, also referred to as ∠X, ∠Y, and ∠Z, respectively.
The vertices are the points where two sides intersect, which in triangle XYZ are X, Y, and Z. These components together define the structure of a triangle.
Perimeter of a Triangle
Properties of Triangle
Triangles have several important properties, including the following, that make them fundamental in geometry:
Sum of Interior Angles
: The sum of the three interior angles of any triangle is always 180 degrees. For example, in triangle XYZ, if ∠X = 60°, ∠Y = 70°, and ∠Z = 50°, their sum equals 180°.
Triangle Inequality Theorem:
The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
For instance
, if a triangle has sides XY, YZ, and ZX, then XY + YZ > ZX.
Side Difference Property
: The difference between the lengths of any two sides of a triangle is always less than the length of the third side. Using the same triangle,
|XY - YZ| < ZX.
Similarity of Triangles:
Two triangles are considered similar if their corresponding angles are equal (congruent) and their corresponding side lengths are proportional. This means that if triangle ABC and triangle XYZ have ∠A = ∠X, ∠B = ∠Y, and ∠C = ∠Z, and the ratio of their sides AB/XY = BC/YZ = CA/ZX, the triangles are similar.
Length, Width, and Height
Types of Triangles
Triangles can be classified in two major ways: based on the length of their sides and based on the measure of their angles. This results in six distinct types of triangles. Let’s explore each classification in detail.
Based on the Length of the Sides:
Scalene Triangle:
In a scalene triangle, all three sides have different lengths. No sides are equal, and all angles are also different from each other.
Isosceles Triangle:
An isosceles triangle has two sides of equal length, and the angles opposite these sides are also equal. The third side is of a different length, making this triangle symmetrical along the axis through the two equal sides.
Equilateral Triangle:
An equilateral triangle has all three sides of equal length. Additionally, all three angles are equal, each measuring 60 degrees, making it also an acute triangle.
Based on the Angles:
Acute Angle Triangle:
An
acute angle triangle has all three angles less than 90 degrees. These triangles are sharp and have no right angles or obtuse angles.
Obtuse Angle Triangle:
An obtuse angle triangle has one angle that is greater than 90 degrees but less than 180 degrees. The other two angles in the triangle will be acute (less than 90 degrees)
Right Angle Triangle:
A right angle triangle has one angle exactly equal to 90 degrees. The side opposite the right angle is the longest side, called the hypotenuse, while the other two sides are the legs of the triangle.
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Angles of Triangle
A triangle has three angles, each formed where two sides meet at a point called the vertex. The sum of these three interior angles is always 180°.
If we extend the sides of the triangle outward, exterior angles are formed. Each interior angle and its adjacent exterior angle add up to 180°.
Let’s say the interior angles are ∠1, ∠2, and ∠3, and their corresponding exterior angles are ∠4, ∠5, and ∠6.
This gives us:
-
∠1 + ∠4 = 180°
-
∠2 + ∠5 = 180°
-
∠3 + ∠6 = 180°
When we add these equations together:
∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 540°
Since ∠1 + ∠2 + ∠3 = 180°, we get:
180° + ∠4 + ∠5 + ∠6 = 540°
This means:
∠4 + ∠5 + ∠6 = 360°
Therefore, the sum of the exterior angles of a triangle is always 360°.
Perimeter of a Triangle
The perimeter
of a triangle is the total length around the triangle, calculated by adding the lengths of all three sides. It represents the boundary of the triangle. The perimeter is expressed in the same unit as the side lengths, whether in centimeters, meters, or any other unit of length.
Perimeter of a Triangle Formula
If a triangle has sides of lengths
AB, BC, and AC, the perimeter (P
) of the triangle is simply the sum of these three side lengths:
Perimeter= AB+ BC+ AC
For example, if a triangle has side lengths of 5 cm, 7 cm, and 8 cm, the perimeter would be:
Perimeter = 5 + 7 + 8 = 20 cm
Area of a Triangle
The
area
of a triangle refers to the amount of space enclosed within its boundaries. The area depends on the triangle’s base and height, and it is measured in square units, such as square centimeters or square meters.
Area of a Triangle Formula (when Base and Height are known):
The area of a triangle can be calculated if we know the
base (B)
and the
height (H)
. The formula for the area is:
Area =½ × Base × Height
This formula works for all types of triangles, provided the base and the corresponding height are known.
Example:
Let’s say we have a triangle with a base of 9 cm and a height of 6 cm. Using the formula:
Area =½ × 9 × 6 = 27cm
2
Thus, the area of the triangle is 27 square centimeters.
Area of a Triangle Using Heron’s Formula
In some cases, the base and height of the triangle are not provided. If we know the lengths of all three sides of the triangle, we can still calculate the area using
Heron’s formula
.
To calculate the area of a triangle using Heron’s formula, we first need to determine the semi-perimeter (s) of the triangle. The semi-perimeter is half of the perimeter and can be calculated as:
s= a + b + c / 2
Here
a
,
b
, and
c
are the lengths of the sides of the triangle.
After calculating the semi-perimeter, the area (
A
) of the triangle can be found using the following formula:
A= s(s−a)(s−b)(s−c)
Where:
-
s
is the semi-perimeter,
-
a
,
b
, and
c
are the sides of the triangle.
Let's understand the
a
bove concept with an example.
Consider a triangle with side lengths 7 cm, 8 cm, and 9 cm.
Step 1:
Calculate the semi-perimeter:
s= 7 + 8 + 9 = 12 cms
Step 2:
Apply Heron’s formula to find the area.
A=12(12−7)(12−8)(12−9)
A=12×60 =720 ≈26.83cm
2
Thus, the area of the triangle is approximately 26.83 square centimeters.
