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Sides of Triangle

The sides of a triangle are straight lines that are joined by the three vertices of the triangle. In other words, we can say that the sides of a triangle are line segments that meet each other at the vertices of the triangle. The sides of a right-angled triangle can be found out by using various methods like the Pythagoras theorem or by using the perimeter of the triangle. In case some of the angles and other side lengths are given, we can use the law of cosines or the law of sines to find the lengths of the sides of a triangle.

In this article, we will explore the concept of the sides of a triangle along with its formula. We will also discuss the properties and rules of the sides of a triangle and solve a few examples based on the concept for a better understanding.

1.	What are the Sides of Triangle?
2.	Sides of Triangle Formula
3.	Sides of Triangle Properties
4.	Sides of Triangle Rule
5.	FAQs on Sides of Triangle

What are the Sides of Triangle?

Each triangle has three sides and three angles. These sides of the triangle are straight line segments such that two sides meet at each vertex of the triangle to form a three-sided closed figure. In a right-angled triangle, each side has a name. The longest side of a right-angled triangle is called the hypotenuse, the lower side of the triangle is called the base and the standing line adjacent to the right angle is called the perpendicular. Three angles are formed at the end of each side of the triangle, that is, at each vertex. Let us now discuss the formula used to find the lengths of these sides.

Sides of Triangle Formula

The sides of a triangle formula of a given triangle to find its sides are related to the trigonometric ratios. The necessary conditions include - one side of the triangle and an acute angle and thus, we can find out the rest of the sides of the triangle.

In the case of a right triangle, we can apply the Pythagorean theorem directly.
In the case of an isosceles triangle, we can use the area or perimeter formula.
In the case of a general, some of the angles and some side lengths are known, we can use the law of cosines or the law of sines.

Sides of a Triangle Formula

Sides of A Triangle Formula

1. If we are given an angle and a side length for a right triangle,

Sine θ = Length of the opposite side / Length of the Hypotenuse side
Cos θ = Length of the adjacent side / Length of the Hypotenuse side
Tan θ = Length of the opposite side / Length of the adjacent side

2. The law of sines: sin(A)/a = sin(B)/b = sin(C)/c

Where,

a, b, and c are the sides of the triangles oppo.
A, B, and C are the corresponding angles.

3. The law of cosines: c² = a² + b² - 2ab cos(C)

Where,

a, b, and c are the sides of the triangles.
C is the angle formed by the sides a and b.

We use the above-given formulas to find the length of the sides of a triangle depending upon the known values of the triangle.

Sides of Triangle Properties

Now that we have discussed the formulas to find the lengths of the sides of a triangle, let us go through some of the important properties about the sides of the triangle:

The side opposite to the greatest angle of the triangle is the longest side of the triangle.
Two triangles are said to be congruent if the lengths of the corresponding sides of the triangle are equal.
Two triangles are said to be similar if the lengths of the corresponding sides of the triangle are proportional.
The sum of three sides of a triangle gives the perimeter of the triangle.
The area of a triangle can be calculated using the three sides of a triangle (Heron's formula) whose formula is:
- Area = √[s(s – a)(s – b)(s – c)], where a, b, c are the three sides of a triangle and s is the semi-perimeter.

Sides of Triangle Rule

So far we have discussed the important properties of the sides of a triangle, let us now understand its basic rule. The rule of the sides of a triangle is that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. This rule is also known as the triangle inequality theorem. This implies that we cannot have a triangle with lengths 3, 4, 9 as 3 + 4 = 7 < 9. Let us see the applications of sides of a triangle formula and rule in the following section.

Important Notes on Sides of Triangle

The sides of a triangle are straight lines that are joined by the three vertices of the triangle.
In the case of a right triangle, we can apply the Pythagorean theorem or trigonometric ratios formula to find the sides.
We can use the law of cosines or the law of sines to find the lengths of sides of a triangle.

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Example 1: What are the sides of the right triangular park whose hypotenuse is 10 in and has a base angle of 30°?

Solution: To find: The sides of a right triangular park.

Given,

Hypotenuse = 10 in.

The base angle = 30 degrees

Using sides of a triangle Formula,

sin θ = Length of opposite side / Length of Hypotenuse side

⇒ sin 30° = x/10 --- (Assuming Length of opposite side = x)

⇒ 1/2 = x/10

⇒ x = 5 in

And, cos θ = Length of adjacent side / Length of Hypotenuse side

⇒ cos 30°= y/10 --- (Assuming Length of adjacent side = y)

⇒ √3/2 = y/10

⇒ y = 5√3 in

Answer: The remaining sides of a triangular park are 5 in and 5√3 in.
Example 2: The lengths of the two sides of a triangle ABC are 10 units and 9 units and the angle between them is 47°. Find the length of the third side of the triangle.

Solution: To find the length of the third side of the triangle, we will use the law of cosines.

We have a = 10, b = 9, and angle C = 47°. We need to find the value of c. So, we have

c² = a² + b² - 2ab cos(C)

⇒ c² = 10² + 9² - 2 × 10 × 9 cos(47°)

= 100 + 81 - 180 × 0.682

= 58.24

⇒ c = √58.24

= 7.63 units.

Answer: The length of the third side of the triangle is 7.63 units.
Example 3: In triangle ABC, ∠C = 42° and ∠A = 33°, and the side opposite to angle C is 12.5 units. Find the length of the side of the triangle opposite to angle A.

Solution: We have ∠C = 42° and ∠A = 33°, c = 12.5 units. We need to find the side 'a'. So, using the sine law, we have

sinA / a = sinC / c

⇒ sin (33°) / a = sin (42°) / 12.5

⇒ 0.545 / a = 0.67 / 12.5

⇒ a = 12.5 × 0.545 ÷ 0.67

= 10.17 units

Answer: The length of the side opposite to angle A is 10.17 units.

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Sides of Triangle Questions

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FAQs on Sides of Triangle

What are the Sides of Triangle in Geometry?

The sides of the triangle are straight line segments such that two sides meet at each vertex of the triangle to form a three-sided closed figure.

What Is the Sides of a Triangle Formula?

To find the sides of a triangle, we use different formulas depending upon the known values for the given triangle. We use the law of cosines or the law of sines if some sides and some angles are given. We can use the Pythagorean theorem to find the sides of the right-angled triangle. We can also use the formula of the trigonometric ratio in the case of a right-angled triangle.

When To Use Sides of a Triangle Formula?

We can use the trigonometric ratios formula or the Pythagorean theorem formula only in the case of a right-angled triangle as it involves trigonometric ratios to be applied to find the sides of the given triangle. In case, some of the angles and other side lengths are given, we can use the law of cosines or the law of sines.

How To Use Sides of a Triangle Formula?

If we are given an angle and a side length of a triangle,

Step 1: Check for the type of triangle.
Step 2: Check for any known sides or angles.
Step 3: Put the given values in the sides of a triangle formula: Sine θ = Length of the opposite side / Length of the Hypotenuse side, Cos θ = Length of the adjacent side / Length of the Hypotenuse side, Tan θ = Length of the opposite side / Length of the adjacent side or directly use the law of sine or law of cosine accordingly.

What Are the Applications of the Sides of a Triangle Formula?

The formula for the sides of a triangle has applications in trigonometry which further has a varied number of applications in our everyday life, from the creation of maps to the construction of buildings.

What are the Important Properties of the Sides of Triangle?

Some of the important properties about the sides of the triangle:

The side opposite to the greatest angle of the triangle is the longest side of the triangle.
Two triangles are said to be congruent if the lengths of the corresponding sides of the triangle are equal.
Two triangles are said to be similar if the lengths of the corresponding sides of the triangle are proportional.
The sum of three sides of a triangle gives the perimeter of the triangle.

What is the Sides of Triangle Rule?

The rule of the sides of a triangle is that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. This rule is also known as the triangle inequality theorem.

What are the 3 Sides of a Right Triangle?

The 3 sides of a right-angled triangle are Hypotenuse (the longest side), Perpendicular (also, called the opposite side), and the Base (also, called the adjacent side).

How to Find the Sides of Triangle with Angles?

To find the length of the side of a triangle with angles, we need at least one side. Then, we can use the law of sines to find the length of the sides of the triangle.

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