If RS is the altitude drawn to the hypotenuse of the right-angled triangle as shown in the above figure, then which of the following are true?

h^{2} = x y

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h^{2} = \frac{x}{y}

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△ P R Q \sim △ P S R

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p^{2} = y r

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Solution

The correct options are
A $h^{2} = x y$
C $△ P R Q \sim △ P S R$
D $p^{2} = y r$
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.
i.e., $△ P R Q \sim △ P S R \sim △ R S Q$
(ii) $h^{2} = x y$
(iii) $p^{2} = y r$ and $q^{2} = x r,$ where $r = x + y .$
The above theorem is called altitude-on-hypotenuse theorem.

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