Altitude-on-Hypotenuse theorem — lesson. Mathematics State Board, Class 8.

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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

$6$

$cm$ .

Did you find an error?

3. Altitude-on-Hypotenuse theorem