# Altitude of scalene triangle

In this chapter, we will derive formula to calculate the length of altitude in scalene triangle.

## Formula for length of altitude – Scalene triangle

The triangle in which all sides are of different length are called scalene triangle.

Given below is the scalene triangle ABC with altitude AM.

Here length of sides is given as;
AB = a
BC = b
CA = c
Altitude AM = h

The area of triangle using Heron’s formula is given as;

\mathtt{Area\ =\ \sqrt{s( s-a)( s-b)( s-c)}}

Where;
s = (a + b + c) / 2

Using altitude, the area of triangle can be written as;

\mathtt{Area\ =\ \frac{1}{2} \times \ base\ \times \ height}\\\ \\ \mathtt{Area\ =\ \ \frac{1}{2} \times \ b\times \ h}

Equating both the area expressions.

\mathtt{\frac{1}{2} \times \ b\times \ h=\ \sqrt{s( s-a)( s-b)( s-c)}}\\\ \\ \mathtt{h\ =\ \frac{2\mathtt{\sqrt{s( s-a)( s-b)( s-c)}}}{b}}

Hence, the length of altitude in scalene triangle can be calculated using above formula.

Please try to remember the formula as it would help us solve question faster.

## Questions on finding altitude of scalene triangle

Example 01
Given below are scalene triangle of length 3 cm, 4 cm and 5 cm. Find the length of altitude AQ.

Solution
Let us first calculate value of s.

\mathtt{s\ =\ \frac{a+b+c}{2}}\\\ \\ \mathtt{s=\ \frac{3+4+5}{2}}\\\ \\ \mathtt{s=\ 6}

The altitude formula for scalene triangle is given as;

\mathtt{h\ =\ \frac{2\mathtt{\sqrt{s( s-a)( s-b)( s-c)}}}{b}}

Putting the values;

\mathtt{h\ =\ \frac{2\mathtt{\sqrt{6( 6-3)( 6-5)( 6-4)}}}{5}}\\\ \\ \mathtt{h\ =\ \frac{2\mathtt{\sqrt{6( 3)( 1)( 2)}}}{5}}\\\ \\ \mathtt{h\ =\ \frac{2\mathtt{\sqrt{36}}}{5}}\\\ \\ \mathtt{h\ =\ \frac{12}{5}}\\\ \\ \mathtt{h\ =\ 2.5\ cm\ }

Hence, length of altitude of 2.5 cm

Example 02
Given below is the scalene triangle with sides 5 cm , 6 cm and 7 cm. Find the length of altitude.

Solution
Since the altitude BQ intersect the side AC at 90 degree, we will consider AC as base.

First calculating value of s.

\mathtt{s=\ \frac{5+6+7}{2}}\\\ \\ \mathtt{s=\ 9}

Now using the altitude formula for scalene triangle.

\mathtt{h\ =\ \frac{2\mathtt{\sqrt{s( s-a)( s-b)( s-c)}}}{b}}

Putting the values;

\mathtt{h\ =\ \frac{2\mathtt{\sqrt{9( 9-5)( 9-6)( 9-7)}}}{6}}\\\ \\ \mathtt{h\ =\ \frac{2\mathtt{\sqrt{9( 4)( 3)( 2)}}}{6}}\\\ \\ \mathtt{h\ =\ \frac{2\mathtt{\sqrt{216}}}{6}}\\\ \\ \mathtt{h\ =\ \frac{2\times 6\sqrt{6}}{6}}\\\ \\ \mathtt{h\ =\ 2\sqrt{6} \ cm\ }

Hence, \mathtt{h\ =\ 2\sqrt{6} \ cm\ \ } is the required solution.