How to prove that the line joinning the midpoints of the diagonals of trapezium is parallel to the parallel sides of the trapezium?

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Solution 1

Perhaps it is not surprising that all the segments (not only those that are actually depicted) going through the midpoint of the red line are halved by the same red point:

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Taking any trapezium the two diagonals will be parallel to two of the colored lines above. That is, their midpoints will be equidistant to both of the black parallel lines...

Solution 2

let $ABCD$ be trapezium $M$ be mid points of diagonals $AC$&$P$ be mid points of $BC$

by mid point theorem

$PM$ is parallel to$AB$

$PM$ intersect$BD$ at $N$

so $PN$ is parallel to $AB\implies $\parallel CD to$CD$

hence by converse of mid point theorem

$N$ is mid point of $BD$

again it meet $DA$ at $Q$

now $NQ$ is parallel to $CD$

by converse again q is mid point of $DA$

hence proved

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Math, math and only math

Updated on August 01, 2022

Comments

  • ami_ba
    ami_ba over 1 year

    How to prove that the line joining the midpoints of the diagonals of trapezium is parallel to the parallel sides of the trapezium?

    I tried to prove this by drawing BD and joining the midpoints of AD and BC. I am unable to prove that this line and the line joining the midpoints of the diagonals are the same, please help

  • ami_ba
    ami_ba almost 7 years
    I found your solution fantastic, thank you vv much!!
  • zoli
    zoli almost 7 years
    Thank you, then what about accepting it?
  • ami_ba
    ami_ba almost 7 years
    Yeah, I accepted it.
  • zoli
    zoli almost 7 years
    THX, We all work for points :)