$ABCD$ is a parallelogram in which $P$ and $Q$ are mid-points of opposite sides $AB$ and $CD$ respectively. If $AQ$ intersects $DP$ at $S$ and $BQ$ intersects $CP$ at $R$,show that $DPBQ$ is a parallelogram.
(N/A) Given that $ABCD$ is a parallelogram,so $AB \,|| \,DC$ and $AB = DC$.
Since $P$ and $Q$ are mid-points of $AB$ and $CD$ respectively,we have $PB = \frac{1}{2} AB$ and $DQ = \frac{1}{2} DC$.
Since $AB = DC$,it follows that $\frac{1}{2} AB = \frac{1}{2} DC$,which implies $PB = DQ$.
Also,since $AB \,|| \,DC$,we have $PB \,|| \,DQ$.
In quadrilateral $DPBQ$,we have one pair of opposite sides ($PB$ and $DQ$) equal and parallel.
Therefore,$DPBQ$ is a parallelogram.
Since $P$ and $Q$ are mid-points of $AB$ and $CD$ respectively,we have $PB = \frac{1}{2} AB$ and $DQ = \frac{1}{2} DC$.
Since $AB = DC$,it follows that $\frac{1}{2} AB = \frac{1}{2} DC$,which implies $PB = DQ$.
Also,since $AB \,|| \,DC$,we have $PB \,|| \,DQ$.
In quadrilateral $DPBQ$,we have one pair of opposite sides ($PB$ and $DQ$) equal and parallel.
Therefore,$DPBQ$ is a parallelogram.
Explore More
Similar Questions
Two parallel lines $l$ and $m$ are intersected by a transversal $p$ (see Fig). Show that the quadrilateral formed by the bisectors of interior angles is a rectangle.
Difficult
View Solution$l, m$ and $n$ are three parallel lines intersected by transversals $p$ and $q$ such that $l, m$ and $n$ cut off equal intercepts $AB$ and $BC$ on $p$ (see Fig). Show that $l, m$ and $n$ cut off equal intercepts $DE$ and $EF$ on $q$ also.
Medium
View SolutionIn $\Delta ABC$ and $\Delta DEF$,$AB = DE$,$AB \parallel DE$,$BC = EF$ and $BC \parallel EF$. Vertices $A, B$ and $C$ are joined to vertices $D, E$ and $F$ respectively (see figure). Show that $AD \parallel CF$ and $AD = CF$.
Medium
View Solution$ABCD$ is a rhombus and $P, Q, R$ and $S$ are mid-points of the sides $AB, BC, CD$ and $DA$ respectively. Show that the quadrilateral $PQRS$ is a rectangle.
Difficult
View SolutionIn $\Delta ABC$,$D$,$E$,and $F$ are respectively the mid-points of sides $AB$,$BC$,and $CA$. Show that $\Delta ABC$ is divided into four congruent triangles by joining $D$,$E$,and $F$.
Medium
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo