Triangle is the simplest form of a Polygon. The word “Tri” means three and therefore a figure with 3 angles is a triangle, and It is formed with the help of three-line segments intersecting each other, a triangle has 3 vertices, 3 edges, and 3 angles. The shape of a triangle is very useful in real life too, like carpentering, Astronomy, street signboards, etc.
A triangle is a polygon with three sides and three angles. It is denoted by the symbol △. In a triangle where any two-side joint is called a vertex. A triangle has three vertices. It is one of the basic figures used in mathematics, there are various properties of the triangle which are discussed below
Angles in a Triangle
A triangle has three angles, an angle is formed when two sides of the triangle meet at a common point, this common point is known as the vertex. Sum of the three interior angles is equal to 180 degrees.
When the sides are extended outwards in a triangle, then it forms three exterior angles. The sum of interior and exterior angles pair of a triangle is always supplementary. Also, the sum of all three exterior angles of a triangle is 360 degrees.
Properties of a Triangle
- Angle Sum Property: The sum of all three interior angles is always 180°. Therefore. In the Triangle ΔABC shown above, ∠A+ ∠B+ ∠C= 180°, interior angles of a triangle will be greater than 0° and less than 180°.
- A Triangle has 3 sides, 3 vertices, and 3 angles.
- Exterior angle property: The Exterior angle of a triangle is equal to the sum of Interior opposite and non-adjacent angles (also referred to as remote interior angles). In the above shown ΔABC, ∠ACD= ∠ABC+ ∠BAC
- The sum of the length of any two sides of a triangle is always greater than the third side. For example, AB+ BC> AC or BC+ AC> AB.
- The side opposite the largest angle is the largest side of the triangle. For instance, in a right-angled triangle, the side opposite 90° is the longest side.
- The perimeter of a figure is defined by the overall length the figure is covering. Hence, the perimeter of a triangle is equal to the sum of lengths on all three sides of the triangle. Perimeter of ΔABC= (AB + BC + AC)
- The difference between the length of any two sides is always lesser than the third side. For example, AB-BC< AC or BC-AC< AB
- For similar triangles, the angles of the two triangles have to be congruent to each other and the respective sides should be proportional.
- Area of Triangle: 1/2× base × height
In geometry, for every two-dimensional shape (2D shape), there are always two basic measurements that we need to find out, i.e., the area and perimeter of that shape. Therefore, the triangle has two basic formulas which help us to determine its area and perimeter. Let us discuss the formulas in detail.
Perimeter of Triangle
Perimeter of a triangle is defined as the sum of all three sides of a triangle. Suppose a triangle with sides a, b, and c is given then its perimeter is given by
Perimeter of triangle = a + b + c
Area of a Triangle
Area of a triangle is the total area covered by the triangle boundary. It is equal to half the product of its base and height. Area of a triangle is measured in square units. If the base of a triangle is b and its height is h then its area is given by
Area of Δ = 1/2 × b × h
Area of Triangle Using Heron’s Formula
When the three sides of a triangle are known but its height is unknown then its area is calculated using Heron’s Formula. Suppose a triangle with sides a, b, and c units is given then its area is calculated using steps discussed below
Steps to Find Area Using Herons Formula
Step 1: Mark all the dimensions of the given triangle.
Step 2: Calculate the semi perimeter (s) using formula s = (a+b+c) / 2
Step 3: Use the formula for finding area Area;
A = √[s(s-a)(s-b)(s-c)]
a,b, and c are the sides of a triangle
Types of Triangles
Classification of triangles is done based on the following characteristics:
- Types of Triangles Based on Sides
- Types of Triangles Based on Angles
Types of Triangles Based on Sides
On the basis of sides, triangles are classified as
- Scalene Triangle
- Isosceles Triangle
- Equilateral Triangle
In an Equilateral triangle, all three sides are equal to each other as well as all three interior angles of the equilateral triangle are equal.
Since all the interior angles are equal and the sum of all the interior angles of a triangle is 180° (one of the Properties of the Triangle). We can calculate the individual angles of an equilateral triangle.
∠A+ ∠B+ ∠C = 180°
∠A = ∠B = ∠C
Therefore, 3∠A = 180°
∠A= 180/3 = 60°
Hence, ∠A = ∠B = ∠C = 60°
Properties of Equilateral Triangle
- All sides are equal.
- All angles are equal and are equal to 60°
- There exist three lines of symmetry in an equilateral triangle
- The angular bisector, altitude, median, and perpendicular line are all same and here it is AE.
- The orthocenter and centroid are the same.
Equilateral Triangle Formulas
The basic formulas for equilateral triangles are:
- Area of Equilateral Triangle = √3/4 × a2
- Perimeter of Equilateral Triangle = 3a
a is the side of triangle
In an Isosceles triangle, two sides are equal and the two angles opposite to the sides are also equal. It can be said that any two sides are always congruent. Area of an Isosceles triangle is calculated by using the formula for area of the triangle as discussed above.
Properties of Isosceles Triangle
- Two sides of the isosceles triangle are always equal
- Third side is referred to as the base of the triangle and the height is calculated from the base to the opposite vertex
- Opposite angles corresponding to the two equal sides are also equal to each other.
In a Scalene triangle, all sides and all angles are unequal. Imagine drawing a triangle randomly and none of its sides are equal, all angles differ from each other too.
Properties of Scalene Triangle
- None of the sides are equal to each other.
- The interior angles of the scalene triangle are all different.
- No line of symmetry exists.
- No point of symmetry can be seen.
- The interior angles may be acute, obtuse, or right angles in nature (this is the classification based on angles).
- The smallest side is opposite the smallest angle and the largest side is opposite the largest angle (general property).
Types of Triangles Based on Angles
On the basis of angles, triangles are classified as
- Acute Angled Triangle
- Obtuse Angled Triangle
- Right Angled Triangle
Acute Angled Triangle
In Acute angle triangles, all the angles are greater than 0° and less than 90°. So, it can be said that all 3 angles are acute in nature (angles are lesser than 90°)
Properties of Acute Angled Triangles
- All the interior angles are always less than 90° with different lengths of their sides.
- The line that goes from the base to the opposite vertex is always perpendicular.
Obtuse Angled Triangle
In an obtuse angle Triangle, one of the 3 sides will always be greater than 90° and since the sum of all three sides is 180°, the rest of the two sides will be less than 90°(angle sum property).
Properties of Obtuse Angled Triangle
- One of the three angles is always greater than 90°.
- The sum of the remaining two angles is always less than 90° (angle sum property).
- The circumference and the orthocenter of the obtuse angle lie outside the triangle.
- The Incenter and the centroid lie inside the triangle.
Right Angled Triangle
When one angle of a triangle is exactly 90°, then the triangle is known as the Right Angle Triangle.
Properties of Right-angled Triangle
- A Right-angled Triangle must have one angle exactly equal to 90°, it may be scalene or isosceles but since one angle has to be 90°, hence, it can never be an equilateral triangle.
- The side opposite 90° is called Hypotenuse.
- The sides adjacent to the 90° are base and perpendicular.
- Pythagoras Theorem: It is a special property for Right-angled triangles. It states that the square of the hypotenuse is equal to the sum of the squares of base and perpendicular i.e. AC2 = AB2 + BC2
Solved Examples on Triangles
Example 1: In the triangle. ∠ACD = 120°, and ∠ABC = 60°. Find the type of Triangle.
In the above figure, we can say, ∠ACD = ∠ABC + ∠BAC (Exterior angle Property)
120° = 60° + ∠BAC
∠BAC = 60°
∠A + ∠B + ∠C = 180°
∠C OR ∠ACB = 60°
Since all the three angles are 60°, the triangle is an Equilateral Triangle.
Example 2: The triangles with sides of 5 cm, 5 cm, and 6 cm are given. Find the area and perimeter of the Triangle.
Given, the sides of a triangle are 5 cm, 5 cm, and 6 cm
Perimeter of the triangle = (5 + 5 + 6) = 16 cm
Semi Perimeter = 16 / 2 = 8 cm
Area of triangle = √s(s – a)(s – b)(s – c) (Using Heron’s Formula)
= √8(8 – 5)(8 – 5)(8 – 6)
= √144 = 12 cm2
Example 3: In the Right-angled triangle, ∠ACB = 60°, and the length of the base is given as 4cm. Find the area of the Triangle.
Using trigonometric formula of tan60°,
tan60° = AB / BC = AB /4
AB = 4√3cm
Area of Triangle ABC = 1/2
= 1/2 × 4 × 4√3
= 8√3 cm2
Example 4: In ΔABC if ∠A+ ∠B = 55°. ∠B + ∠C = 150°, Find angle B separately.
Angle Sum Property of a Triangle says ∠A+ ∠B+ ∠C= 180°
∠A+ ∠B = 55°
∠B+ ∠C = 150°
Adding the above 2 equations,
∠A+ ∠B+ ∠B+ ∠C= 205°
180°+ ∠B= 205°
∠B = 25°
FAQs on Triangles
Question 1: What is a triangle?
A triangle is a polygon, with three sides and three vertices. A triangle also has three angles and the sum of all three angles of the triangle is 180 degrees.
Question 2: What are the main triangle formulas?
The main triangle formulas are,
Area of triangle
A = [(½) b × h]
where ‘b’ is the base and ‘h’ is the height of the triangle.
Perimeter of a triangle
P = (a + b + c)
where ‘a’, ‘b’, and ‘c’ are the sides of the triangle.
Question 3: What are the six basic types of triangles?
The six basic types of triangles are:
- Acute angle triangles
- Obtuse angle triangles
- Right angle triangles
- Scalene Triangles
- Isosceles triangles
- Equilateral triangles
Question 4: What are the properties of triangles?
Some of the important properties of triangles are:
- Sum of all three angles of the triangle is equal to 180 degrees.
- Equal angles have equal sides opposite them.
- Sum of the length of any two sides of a triangle is always greater than the third side.
Question 5: Is an Isosceles triangle always an acute angle triangle?
No, an isosceles triangle is not always an acute angle triangle it can be an acute angle, right angle, or obtuse-angled triangle depending upon the value of the angles of a triangle.
Question 6: What is the difference between scalene, isosceles, and equilateral triangles?
On the basis of sides triangles are of three types:
- Scalene Triangle: A triangle with all three sides unequal is called a Scalene triangle.
- Isosceles Triangle: A triangle in which any two sides are equal is an Isosceles triangle.
- Equilateral Triangle: A triangle with all three sides equal is called an Equilateral triangle.
Question 7: What is the difference between an acute angle triangle, an obtuse angle triangle, and a right angle triangle?
On the basis of angles triangles are of three types:
- Acute Angle Triangle: A triangle with all three angles as acute angles is called an acute angle triangle.
- Obtuse Angle Triangle: A triangle with any one angle as an obtuse angle is called an obtuse angle triangle.
- Right Angle Triangle: A triangle with any one angle as a right angle is called a right angle triangle.
Question 8: Explain why a Right-angled triangle can never be an equilateral triangle.
A Right angled triangle is one with any one of its angles equal to 90°, and other angles are less than 90°. Whereas in an equilateral triangle, all the interior angles are equal and are equal to 60° thus, it can never be possible in a right-angle triangle. So a right-angle triangle can never be an equilateral triangle.
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