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### 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:

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 $$6$$ $$cm$$.