Question

In the right-angled triangle QPR given below, PS is the altitude to the hypotenuse. The figure is followed by three possible inferences.
I.Triangle PQS and Triangle RPS are similar.
II.Triangle PSQ and Triangle RSP are congruent.
III. Triangle PSQ and triangle RPQ are similar.

• A. I and II are correct
• B. I and III are correct
• C. Only II is correct
• D. All three are correct

Answer 1

The correct option is B I and III are correct

Let $\angle PQR=\theta$

$⟹\angle PRQ=\angle QPS={90}^{\circ }-\theta$

$⟹\angle SPR={90}^{\circ }-({90}^{\circ }-\theta )=\theta$

Since the corresponding angles of the $△sPQS,RPS$ and $RQP$ are equal. Hence statement (I) and (III) are correct

However, the sides might not be the same. Hence, statement (II) is not correct