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A triangle is a polygon, having three sides, edges, and vertices. In geometry, the sides of a triangle form the basic shape. Three sides, three angles, and three vertices make up a triangle. For example, In the triangle ABC is represented as ABC, the three sides are AB, BC, CA, the three angles are A, B, C, and the three vertices are A, B, C. The inner angles of any triangle, which are 180 degrees, are formed by the sum of three angles. A median is a line segment that connects the vertex of a triangle to the center point on opposing sides.
Read Also: NCERT Solution Chapter 7 Triangles
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Key Terms: Scalene triangle, Isosceles triangle, Equilateral triangle, Pythagoras theorem, Perimeter formula
Types of Triangles
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There are several ways in geometry to find the sides of a triangle, such as the Pythagoras theorem, Sine and Cosine rule, or the angle sum property of a triangle. These approaches can be used depending on the conditions or parameters provided. The length of a triangle's sides determines its classification. In general, there are three sorts of triangles based on the triangle's sides:
Scalene Triangle: It is a triangle with uneven sides.
Isosceles Triangle: Only two sides of an isosceles triangle are equal, and the angles opposite the equal sides are likewise equal.
Equilateral Triangle: A triangle in which all three sides have the same length and all angles are 60 degrees.
Types of Triangles
Check Important MCQs for Triangles
How to Find the Sides of a Triangle
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Sides of a triangle can be found using various methods which are given below:
Pythagoras Theorem
One of the methods to find the sides of a triangle is the Pythagoras theorem. If the hypotenuse, perpendicular, and base are the sides of a right triangle, then the square of the hypotenuse side is equal to the sum of the squares of the base and perpendicular, according to the theorem.
Base2 + Perpendicular2 = Hypotenuse2
As a result, the third side of the triangle can be found easily if any two sides are known.
Finding the Length of the Triangle Given One Side and Angle
Trigonometric ratios can be utilized to calculate the other two sides if an angle and a side length are known. According to the sine, cosine, and tangent ratios, if θ is the angle between two sides in a triangle, then.
\(\text{Sine }\theta = \frac{\text{Opposing side length}}{\text{Hypotenuse side length}}\)
\(\text{Cos }\theta = \frac{\text{Base side length}}{\text{Hypotenuse side length}}\)
\(\text{Tan }\theta = \frac{\text{Perpendicular side length}}{\text{Base side length}}\)
Using Perimeter Formula
Any triangle's perimeter is equal to the sum of its sides. A Perimeter is any triangle's entire length. If a triangle ABC is given, then use the formula to solve it.
\(\text{Perimeter of }\triangle \text{ABC} = \text{AB + BC + AC}\)
The length of the third side can be easily calculated if the length of any two sides and the perimeter of the triangle are known.
Also Check: Properties of Triangles
Things to Remember
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- A Scalene Triangle is a triangle with uneven sides.
- An isosceles triangle has just two equal sides, and the angles opposite the equal sides are also equal.
- With the help of Pythagoras theorem, the third side can be found if the two sides are known.
- The third side of the triangle can also be found with help of the perimeter of the given triangle. The perimeter of the triangle is the sum of all the given sides.
- If an angle and a side length are known, trigonometric ratios can be used to derive the other two sides. The sine and cosine rules are used to describe this.
Read More: Euclidean Geometry
Sample Questions
Ques. An isosceles triangle has a perimeter of 100 cm. Find the length of the equal sides if the base is 36 cm. (3 marks)
Ans. An isosceles triangle has two equal sides, both denoted by x
Perimeter = x + x + 36 = 100
2x = 64
x = 32
Hence, the length of the equal sides is 32 cm.
Ques. The perimeter of a triangle ABC is 150 cm, while the two sides AB and BC are 50 and 60 cm long, respectively. Then figure out how long the third side is. (3 marks)
Ans. Perimeter = AB + BC + AC
150 = 50 + 60 + AC
AC = 150 – 110
AC = 40 cm
Hence, the length of the third side of the triangle is 40 cm.
Ques. Find the length of the perpendicular in a right triangle with a base of 4cm and a hypotenuse of 5cm. (3 marks)
Ans. According to Pythagoras theorem,
Hypotenuse2 = Base2 + Perpendicular2
c2 = a2 + b2
b2 = c2 – a2
b2 = 52 – 42 = 25 – 16 = 9
b = √9 = 3
Hence, the length of the perpendicular is 3 cm.
Ques. Find the third angle of a triangle if the first two angles are 45° and 60°, respectively. (3 marks)
Ans. Given, angle A= 45º, angle B= 60º
Using angle sum property
Angle A + angle B + angle C = 180º
Angle C = 180º - (angle A + angle B)
Angle C = 75º
Hence, the third angle of the given triangle is 75º
Ques. Line l is the perpendicular bisector of AB, which is a line segment. Show that P is equidistant from A and B if it is located on l. (3 marks)
Ans. The line l perpendicular to AB crosses through C, which is AB's midpoint.
To show that PA = PB. Consider the PCA and PCB.
As, AC = BC (C being the midpoint)
angle PCA = angle PCB = 90º
PC is common
So PCA \(\cong\) PCB. through SAS rule
Hence, PA = PB as they are the corresponding sides of the two congruent triangles.
Ques. The base of a triangle is 4 centimeters shorter than the height of the triangle. The triangle's area is 96 cm2. The base has a length of. (3 marks)
Ans. Area= ½ (base)(height)
Given area= 96 cm2
base- height= 4
height(height-4) = 192
height= 16
base= 12
Hence, the base is 12 cm and the height of the triangle is 16 cm.
Ques. The area of a triangle is 615 cm2. What is the length of the perpendicular that is dropped on that side from the opposite vertex if one of its sides is 123 cm? (3 marks)
Ans. Area= ½ (base)(height)
Area= 615 cm
Base= 123 cm
½ (123) (height) = 615
Height= 10 cm
Hence, the height of the given triangle is 10 cm.
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