In a convex quadrilateral $PQRS$, the areas of triangles $PQS$, $QRS$ and $PQR$ are in the ratio $3:4:1$. A line through $Q$ cuts $PR$ at $A$ and $RS$ at $B$ such that $PA:PR=RB:RS$. Prove that $A$ is the midpoint of $PR$ and $B$ is the midpoint of $RS$.
Problem
Source: NMTC 2019 junior P1
Tags: geometry
Leo142857
02.11.2019 17:11
Are you sure there isn’t a typo with the question? Cuz if A is midpoint of PR and B is midpoint of RS wouldn’t it be RA:PR= RB:RS, not PA:PR= RB:RS?
Purple_Planet
02.11.2019 17:35
Leo142857 wrote: Are you sure there isn’t a typo with the question? Cuz if A is midpoint of PR and B is midpoint of RS wouldn’t it be RA:PR= RB:RS, not PA:PR= RB:RS? The problem is typed correctly...
Purple_Planet
03.11.2019 06:56
Purple_Planet wrote: In a convex quadrilateral $PQRS$, the areas of triangles $PQS$, $QRS$ and $PQR$ are in the ratio $3:4:1$. A line through $Q$ cuts $PR$ at $A$ and $RS$ at $B$ such that $PA:PR=RB:RS$. Prove that $A$ is the midpoint of $PR$ and $B$ is the midpoint of $RS$. Anyone???
Leo142857
03.11.2019 07:34
Hint: use ratios of triangles’ areas by their bases
Leo142857
03.11.2019 10:55
https://drive.google.com/file/d/1zgmeFe7mq4CL4-DVYNvhufylcCsAAQHX/view