question_answer

A point X inside a rectangle PQRS is joined to the vertices then, which of the following is true?

A)  area\[\left( \Delta PSX \right)=area\left( \Delta PXQ \right)\]

B)  area\[\left( \Delta PSX \right)\]+area\[\left( \Delta PXQ \right)\]=area(RSX) area\[\left( \Delta RQX \right)\]

C)  area\[\left( \Delta PXS \right)\]+area\[\left( \Delta RXQ \right)\]=area\[\left( \Delta SRX \right)\] +area\[\left( \Delta PXQ \right)\]

D)  None of these

Correct Answer: C

Solution :

(c): Draw perpendicular from 'X' \[\left( Let\text{ }PS=RQ=a;\text{ }PQ=RS=b \right)\] Area \[\left( \Delta PSX \right)\] + Area \[\left( \Delta RXQ \right)\] \[=\frac{1}{2}{{x}_{1}}.a+\frac{1}{2}.{{x}_{2}}.a\] \[=\frac{a}{2}({{x}_{1}}+{{x}_{2}})~\frac{a.b}{2}\] Area (PXQ) + Area (RXS) \[=\frac{1}{2}.{{x}_{3}}.b+\frac{1}{2}.{{x}_{4}}.b\] \[=\frac{b}{2}({{x}_{3}}+{{x}_{4}})=\frac{ab}{2}(as,{{x}_{3}}+{{x}_{4}}=a)\] Option (c) is true;