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In the geometry of mathematics, we come across different types of shapes – some two dimensional and some three dimensional. Shapes that can be defined with the help of two parameters are called two-dimensional shapes and those which can be defined with three parameters form three-dimensional shapes. Two-dimensional shapes include squares, rectangles, triangles, etc. A polygon is such a two-dimensional structure having straight sides. A triangle consists of three line segments placed at a particular angle for any pair of the three line segments. A triangle has three angles whose sum is constant which is 180°.
Let a triangle be △PQR.
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The sum of the angles of △PQR is 180°.
Hence, ∠P + ∠Q + ∠R = 180°.
Types of Triangle
Based on side lengths of a triangle, triangles can be classified as-
Triangles in which all the sides of the triangle are equal in length.
Triangles in which any two sides of the triangle are equal resulting in equal base angles.
Triangles in which none of the sides has an equal length or all the sides of the triangle are different in lengths.
Based on the interior angles of a triangle, triangles can be classified as –
Triangles in which all the angles are less than 90°.
Triangles in which any one of the angles is exactly 90°.
Triangle in which any one of the angles is more than 90°.
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Triangle Formula
If ∠A, ∠B, and ∠C are the interior angles of a triangle, ∠A + ∠B + ∠C = 180°.
If b denotes the base of a triangle and h denotes the altitude of the triangle, the area of the triangle is given 12 х base х height square units.
If a,b and c are the sides of a triangle, the area of the triangle can be given by Heron’s formula = [sqrt{s(s – a)(s – b)(s – c)}]
(where s stands for semi perimeter which is 12(a+b+c)
If side lengths of a triangle are denoted by a, b and c, then the perimeter of the triangle is given by (a+b+c).
In a right-angled triangle, if p denotes perpendicular, b denotes base, and h denotes hypotenuse, h2= b2+ p2. This is known as Pythagoras’ Theorem.
Solved Examples
1. Calculate the area of a right-angled triangle in which the measurement of the hypotenuse is 5cm and that of the base is 3cm.
Ans: Given, base = 3cm, hypotenuse = 5cm.
We are given a right-angled triangle whose hypotenuse and base are known.
To find out the area of a triangle, we either need to know the values of base and height or the lengths of all the sides of the triangle. To find out the perpendicular of the triangle, we may use the Pythagoras theorem. The perpendicular of a right-angled triangle corresponds to the altitude of a triangle.
Let altitude/perpendicular be p cm.
For a right – angled triangle,
By Pythagoras Theorem,
Perpendicular2 = hypotenuse2 – base2
Perpendicular2 = 52 – 32
= 25 – 9
Perpendicular2 = 16
perpendicular = 4
[text{Area of the Triangle} = frac{1}{2} times text{base} times text{height} = frac{1}{2} times 3 times 4 = 6]
Hence, the area of the triangle is 6 cm.
2. The perpendicular of a right-angled triangle is 8 units. The area of the triangle is given to be 24 square units. Find the base, hypotenuse, and perimeter of the triangle.
Ans: Given, perpendicular = 8 units, area = 24 square units
We can find out the base of the triangle from the area formula and hypotenuse by Pythagoras Theorem consequently.
We know,
[Area = frac{1}{2} times base times height]
[base = frac{2 times Area}{height}]
So,
[= frac{2 times 24}{8} = 6]
Now we know the base and perpendicular of the triangle.
By Pythagoras theorem,
Perpendicular2 = hypotenuse2 – base2
Perpendicular2 = 82 – 62
= 64 – 36
Perpendicular2 = 100
perpendicular = 10
Now we know all the three sides of the triangle.
Perimeter = hypotenuse + base + perpendicular
=> Perimeter = 10 + 6 + 8 units
=> Perimeter = 24 units.
Hence, the base is 6 units, the hypotenuse is 10 units and the perimeter of the triangle is 24 units.
Points To Remember
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A triangle is a three shaped polygon with three sides, and the two sides joined on the ends is known as the vertex.
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An angle is formed between the two sides.
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Triangle is a closed two-dimensional shape and all the sides are made of straight lines.
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The point where two straight lines join is known as the vertex. Therefore, the triangle has three vertices with each vertex forming an angle.
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The properties of triangles distinguish them from other shapes.
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Triangles have three angles as well as three sides.
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180 degrees is the sum total of the angles of the triangle.
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360 degrees is the sum of exterior angles of a triangle.
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The interior and exterior angles have a supplementary sum.
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The perimeter of a triangle is the length of the boundary of the triangle on the outside.
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The region occupied by the triangle in a two-dimensional space is known as Area.
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Isosceles, equilateral and Scalene triangle
s are the kinds of triangles that are different from one another on the basis of the length of their sides. -
In the scalene triangle, all three sides are different in length.
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In an Isosceles triangle, any two sides are equal in length.
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In an Equilateral triangle, all three sides are equal in length.
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In an acute triangle, all the angles present in it are less than 90 degrees.
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In an obtuse triangle, any one of the angles is greater than 90 degrees.
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In a right-angled triangle, one angle is equal to 90 degrees.
Tips To Master Triangles
Students must pay closer attention to the rules of triangles in order to solve the sums more effectively and within less time. Calculating the angles and sides can be time-consuming so a student must be well versed with the rules and criteria.
Maths is a subject that requires constant practice and so does this chapter. In order to solve the sums like an expert, a student must practice the sums regularly to stay in touch with the concepts and rules of the triangle.
A student should solve the questions from the textbook in order to prepare for the exam. Practising sums of triangles from RD Sharma can give an idea of possible questions to the students.
The solved examples establish the fundamental knowledge and idea about triangles for the students. It can be helpful in understanding the basic concepts of the chapter including the rules and types of triangles.
Paying attention to the teacher and the classes can help a student understand the chapter in the best possible way. A student must be willing to listen and learn carefully about triangles in order to solve the sums and calculate the area, sides and perimeter of the triangle.
A student must make notes simultaneously during the classes so that the notes can help later during revision sessions. It is very important to note the types and the formulas in order to calculate the sums correctly.
A student must be willing to solve the sample papers after solving the examples and questions from the textbook in order to truly master the chapter. Triangles can have various possible questions for exams so it is always better to be prepared.
After solving questions from the books, students may test their knowledge through quizzes which will help them clock their time taken for solving one sum. It is a great way to challenge their skills on a lighter note.
Students can participate in online classes in order to clear their doubts as well as for the revision of this chapter. It will surely help the students understand types of triangles along with their formulas in more efficient ways. It will give them room to make notes and answer the questions to test their skills.
Students can access the online materials such as notes, practice papers, sample questions, solutions to RD Sharma book along with puzzles and quizzes on Congruent Triangles from ’s official website or mobile application. The solutions can be downloaded in PDF formats as well. If a student is stuck at any point in Congruent Triangles then the student can access the online live classes and doubt clearing sessions in order to progress in the chapter.