Question

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

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

Answer 1

The correct option is B I and III are correct

$Statement1:△PQShasangles\angle P=\theta ,\angle Q=90-\theta and\angle S=90△RPShasangles\angle R=\theta ,P=90-\theta and\angle S=90Asallthecorrespondinganglesareequal,△PQS\sim △RPSHence,statement1iscorrect.Statement2:△PSQ,angles\angle P=\theta ,\angle S=90and\angle Q=90-\theta △RSP,angles\angle R=\theta ,\angle S=90,\angle P=90-\theta \Rightarrow △PSQ\sim △RSPbut△PSQ≆△RSPsimilarbutnotcongruent.Hence,2isincorrect.Statement3:△PSQ,angles\angle P=\theta ,\angle S=90and\angle Q=90-\theta △RPQ,angles\angle R=\theta ,\angle P=90and\angle Q=90-\theta Hence,statement3iscorrect.Hence,statement\left(1\right)and\left(3\right)arecorrect.Henceoption\left(B\right)iscorrect.$