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

## Deriving Isosceles altitude formula

The triangle in which two sides are of equal length are called isosceles triangle.

Here length of sides are given as;

AB = a cm

AC = a cm

BC = b cm

Note that in Isosceles triangle, the altitude divides the base into two equal parts.

So, BM = MC = b/2

Now applying Pythagoras theorem in triangle ABM.

\mathtt{AB^{2} =\ BM^{2} +AM^{2}}

Putting the values, we get;

\mathtt{a^{2} =\ \left(\frac{b}{2}\right)^{2} +h^{2}}\\\ \\ \mathtt{h^{2} =\ a^{2} -\frac{b^{2}}{4}}\\\ \\ \mathtt{h^{2} =\ \frac{4a^{2} -b^{2}}{4}}\\\ \\ \mathtt{h=\sqrt{\frac{4a^{2} -b^{2}}{4}}}\\\ \\ \mathtt{h\ =\ \frac{1}{2}\sqrt{4a^{2} -b^{2}}}

Using the above formula, we can calculate the length of the altitude of isosceles triangle.

You have to remember this formula for examination purpose.

## Calculating Isosceles triangle altitude – Solved Problems

**Example 01**

Given below is the Isosceles triangle ABC with sides AB = AC = 4 cm and side BC = 5 cm. Find the length of altitude h.

**Solution**

The formula for altitude of isosceles triangle is given as;

\mathtt{h\ =\ \frac{1}{2}\sqrt{4a^{2} -b^{2}}}

Putting the values;

\mathtt{h\ =\ \frac{1}{2}\sqrt{4( 4)^{2} -5^{2}}}\\\ \\ \mathtt{h\ =\ \frac{1}{2}\sqrt{64-25}}\\\ \\ \mathtt{h\ =\ \frac{1}{2}\sqrt{39}}

Hence, the length of altitude is \mathtt{h\ =\ \frac{1}{2}\sqrt{39}} cm.

**Example 02**

Given below is the isosceles triangle. Find the length of altitude BQ.

**Solution**

Since altitude BQ intersect side AC, we will consider AC as a base of triangle.

Formula for altitude of isosceles triangle is given as;

\mathtt{h\ =\ \frac{1}{2}\sqrt{4a^{2} -b^{2}}}

Putting the values;

\mathtt{h\ =\ \frac{1}{2}\sqrt{4( 6)^{2} -4^{2}}}\\\ \\ \mathtt{h\ =\ \frac{1}{2}\sqrt{144-16}}\\\ \\ \mathtt{h\ =\ \frac{1}{2}\sqrt{128}}\\\ \\ \mathtt{h\ =\ \frac{8\sqrt{2}}{2}}\\\ \\ \mathtt{h\ =\ 4\sqrt{2} \ cm\ }

Hence, \mathtt{4\sqrt{2} \ cm} is the length of required altitude.