### Theory:

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*Illustration*Consider a triangle \(ABC\) right angled at \(A\) with its hypotenuse \(BC\) at its base.

Draw an altitude to the triangle as follows:

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 \(h\).

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 \(6\) \(cm\).