# Scalene Triangle

Triangle is a three sided-polygon with three edges, three vertices and three interior angles. In other words, a triangle is a closed two-dimensional figure with three sides and three angles.

Triangles are classified into different types on the basis of their sides and angles. Each of them has its own individual properties.

Classification of Triangles on the Basis of Their Sides is as Follows:

1. Equilateral Triangle: A triangle whose all the three sides are of equal length.

2. Isosceles Triangle: A triangle whose two sides are of equal length.

3. Scalene Triangle:  A triangle whose all the three sides are of unequal length.

Classification of Triangles on the Basis of Their Angles is as Follows:

1. Acute Angled Triangle: A triangle whose all interior angles are less than 90°.

2. Right Angled Triangle: A triangle whose one of the interior angles is 90°.

3. Obtuse Angled Triangle: A triangle whose one of the interior angles is more than 90°.

### Definition

A scalene triangle is defined as a triangle whose all the three sides are of unequal length and all the three angles are of different measures. However, the sum of all the three interior angles is always equal to 180° degrees.

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In the above figure, the triangle ABC is a scalene triangle, as each of the three sides, AB = 7cm, BC = 9cm and AC = 4cm, are of unequal length.

### Perimeter of a Scalene Triangle

The perimeter of a scalene triangle is the sum of the length of its sides. So, if the length of the sides of the scalene triangle are a, b and c units, then its perimeter is given by:

Perimeter of a scalene triangle = (a + b + c) units

### Area of a Scalene Triangle

The area of a scalene triangle is the amount of space that it occupies in a two-dimensional surface. So, the area of a scalene triangle can be calculated if the length of its base and corresponding altitude (height) is known or the length of its three sides is known or length of two sides and angle between them is given.

Hence, the area of a scalene triangle = $\frac{1}{2}$ × (base) × (height) sq. units

= $\frac{1}{2}$ × (b) × (h) sq. units

Or,

The area of scalene triangle using Heron’s formula = $\sqrt{s(s-a)(s-b)(s-c)}$ sq. units

Where, ‘a’, ‘b’ and ‘c’ are the length of sides of the scalene triangle

And, s = semi-perimeter of triangle = $\frac{a+b+c}{2}$

Or,

The area of the scalene triangle if the length of its two sides and angle between them is given.

Area of scalene triangle = $\frac{1}{2}$ × a × b × sinC

where ‘a’ and ‘b’ are the length of  two sides and C is the angle between them.

### Properties of a Scalene Triangle

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1. It has three sides of unequal length.

2. All the three angles are of different measures.

3. The interior angles of the scalene triangle can be an acute, obtuse or right angle. So, it can be acute-angled or obtuse-angled or right-angled.

4. It has no line of symmetry. So, it cannot be divided into two identical halves.

5. The angle opposite to the longest side of the triangle will be the greatest angle and vice versa.

### Solved Examples:

Q.1. The lengths of sides of a triangular field are  16m, 25m and 35m. Find the cost of fencing the boundary of the field at a cost of ₹15 per meter.

Solution: Given, the side lengths of the triangular field are 16m, 25m and 35m respectively.

The length of the boundary of the triangular field can be calculated using the formula of the perimeter of the triangle.

So, The length of the boundary of the triangular field = 16m + 25m + 35m

= 76m

Now, it is given that the cost of fencing 1-meter boundary is ₹15.

Therefore, the cost of fencing 76m boundary length = 76 x 15 = ₹1140.

Q.2. Find the area of a scalene triangle whose sides measure are 5cm, 8cm and 11cm respectively.

Solution: Let the three given sides be a = 5cm, b = 8cm and c = 11cm.

So, s = semi-perimeter of triangle = $\frac{a+b+c}{2}$ = $\frac{5+8+11}{2}$ =$\frac{24}{2}$ = 12cm

Now, area of the scalene triangle can be calculated using Heron's formula. $\sqrt{12(7)(4)(1)}$

The area of scalene triangle = $\sqrt{s(s-a)(s-b)(s-c)}$ sq. units

Where, ‘a’, ‘b’ and ‘c’ are the length of sides of the scalene triangle and s = semi-perimeter of triangle

On putting the respective values in formula of area of the scalene triangle we get:

The area of scalene triangle = $\sqrt{12(12-5)(12-8)(12-11)}$ sq. cms

= $\sqrt{12(7)(4)(1)}$ sq. cms

= 4 $\sqrt{7Χ3}$ sq. cms

= 4 $\sqrt 21$ sq. cms

= 18.33 sq. cms