Theory:

Illustration:
 
Consider a triangle \(ABC\) right angled at \(A\) with its hypotenuse \(BC\) at its base.
 
Draw an altitude to the triangle as follows:
 
A-H-Theorem.png
 
Two smaller right triangles \(ABD\) and \(ACD\) are obtained.
 
Now, all the three triangles \(ABC\), \(ADB\) and \(ADC\) are similar.
 
Based on this similarity, the following theorem is obtained.
 
Statement:
If an altitude is drawn to the hypotenuse of an right angled triangle, then:

(i) The two triangles are similar to the given triangle and also to each other.
 
That is, \(\Delta ABC \sim \Delta ADB \sim \Delta ADC\).
 
(ii) \(x^2 = yz\)
 
(iii) \(b^2 = za\) and \(c^2 = ya\) where \(a = y + z\)
Example:
From the figure, find the altitude \(h\).
 
A-H-Theorem eg.png
 
Solution:
 
By the statement (ii) of the Altitude-on-Hypotenuse theorem, the altitude is computed as follows:
 
\(h^2 = BD \times DC\)
 
\(h^2 = 4 \times 9\)
 
\(h^2 =\) \(36\)
 
\(\Rightarrow h = \sqrt{36}\)
 
\(= 6\)
 
Therefore, the measure of the altitude is \(6\) \(cm\).