How to Find the Height of a Scalene Triangle

Scalene Triangle:
A triangle in which each side of different length and different angles.


Scalene Triangle

In the given figure AB#BC#CA,therefore the triangle ABC is an scalene triangle. Th three sides measures of different length.

Properties of Scalene Triangle
No sides are symmetry
The angles are not equal 

Types of Scalene Triangle:
Right angled scalene triangle: The triangle in which one of the angle is right angle.
Obtuse angled scalene triangle: The triangle in which one of the angle is obtuse angle.
Acute Scalene Triangle: The three angles of triangle are acute and three sides are of different lengths.

Perimeter of  Triangle:
Sum of the sides of the triangle gives Perimeter of Scalene triangle.
AB + BC + CA = Perimeter

Perimeter of Triangle

Area of Scalene Triangle:
Area = $\frac{1}{2}$ base* height
Length of any side of the given triangle is taken as base and the corresponding altitude is named as height.
When length of the three sides are given are of triangle is given by  Heron's formula
Area = $\sqrt{s(s-a)(s-b)(s-c)}$
where a,b and c  are the lengths of the sides of the triangle

Height of Scalene Triangle:
The height of scalene triangle can be determined using Pythagoras theorem. The height of a triangle is the perpendicular distance to the base from the vertex to opposite the base.
Example: To find the height of scalene triangle whose base is 21m, longest side is 20m and shortest side is 13m.

Solution: Given the base of triangle is 21m, longest side is 20m and shortest side is 13m.

Height of Scalene Triangle

The triangle is divided into two, use Pythagoras theorem to find height of the triangle.
From the figure triangle ABD is given as:
AB2 = BD+ AD2
(20)2 = (21-x)+ h2
(400) = (21)+ (x)2 - 2*21*x + h2
400 = 441 + x2 - 42x + h2    --------(1)

Smaller triangle ADC is given as:
AC2 = AD+ DC2
(13)2 = h2 + x2
(13)2  - x= h2 ---------(2)
assign the value of h in equation (1)
400 = 441 + x2 - 42x + (13)2  - x2                       // simplify   //
400 = 441 - 42x + 169
400 = 610 - 42x
400 - 610 = -42 x
 -210 = -42x
   5 = x

Assign the value of x in equation 2
(13)2  - 52 = h2
169 - 25 =  h2
144  =  h2
12 = h
height of scalene triangle is 12 m