question_answer

\[\Delta PQR\] is a right angled at Q. If X and Y are the mid-points of the sides PQ and QR respectively, then which one of the following is not correct?

A)  \[R{{X}^{2}}+P{{Y}^{2}}=5X{{Y}^{2}}\]

B)  \[R{{X}^{2}}+P{{Y}^{2}}=X{{Y}^{2}}+P{{R}^{2}}\]

C)  \[4\,\,(R{{X}^{2}}+P{{Y}^{2}})=5P{{R}^{2}}\]

D)  \[R{{X}^{2}}+P{{Y}^{2}}=3\,\,(P{{Q}^{2}}+Q{{R}^{2}})\]

Correct Answer: C

Solution :

In \[\Delta PQY,\] \[P{{Y}^{2}}=P{{Q}^{2}}+Q{{Y}^{2}}\] \[\Rightarrow \]\[P{{Y}^{2}}=P{{Q}^{2}}+{{\left( \frac{QR}{2} \right)}^{2}}\]                       ?(i) and in \[\Delta XQR,\] \[R{{X}^{2}}=Q{{X}^{2}}+Q{{R}^{2}}\] \[\Rightarrow \]\[R{{X}^{2}}={{\left( \frac{PQ}{2} \right)}^{2}}+Q{{R}^{2}}\]                       ?(ii) On adding Eqs. (i) and (ii), we get \[P{{Y}^{2}}+R{{X}^{2}}=\frac{5P{{Q}^{2}}}{4}+\frac{5Q{{R}^{2}}}{4}\] \[\Rightarrow \]   \[4\,\,(P{{Y}^{2}}+R{{X}^{2}})=5\,\,(P{{R}^{2}})\]