# prove that the straight line joining the midpoints of the diagonals of a trapezium is parallel to the parallel sides of the trapezium and is equal to half their difference

• 6

Join DE and produce to meet AB at G

In triangle AEG and CEG

<1 = <2  (Alternate interior angles)

AE = EC  (E is the mid point of AC)

<4 = <3  ( Vertically opposite angles)

Triangle AEG is congruent to triangle CED  (by ASA congruence rule)

DE = EG ( CPCT)

AG = CD ( CPCT)

In triangle BGD

E is the mid point of DG

F is the mid point of BD

By mid point theorem,

EF II BG

EF II AB

AB II CD

Hence, AB II EF II CD

EF = 1/2 BG

EF = 1/2 ( AB - AG)

EF = 1/2 (AB - CD)

HOPE THIS HELPS  :)

• 30

@seema.jmu

There's an error in the second line - CEG.

Instead it would be CED (although you have corrected it in the 6th line)

• -4

Ya... it's right.

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• -6

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You can be a master
Don't wait for luck
Dedicate yourself and you can find yourself

Standing in the hall of fame
And the world's gonna know your name
Cause you burn with the brightest flame
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And you'll be on the walls of the hall of fame

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Never gonna know if you never even try

Do it for your counrty
Do it for you name
Cause there's gonna be a day

When your, standing in the hall of fame
And the world's gonna know your name
Cause you burn with the brightest flame
And the world's gonna know your name
And you'll be on the walls of the hall of fame

Be a champion, Be a champion, Be a champion, Be a champion

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Be true seekers

Be students
Be teachers
Be politicians
Be preachers

Be believers
Be astronauts
Be champions

Standing in the hall of fame
And the world's gonna know your name
Cause you burn with the brightest flame
And the world's gonna know your name
And you'll be on the walls of the hall of fame

(You can be a champion)
You could be the greatest
(You can be a champion)
You can be the best
(You can be a champion)
You can be the king kong banging on your chest

(You can be a champion)
You could beat the world
(You can be a champion)
You could beat the war
(You can be a champion)
You could talk to God, go banging on his door

(You can be a champion)
You can throw your hands up
(You can be a champion)
You can be the clock
(You can be a champion)
You can move a mountain
(You can be a champion)
You can break rocks

(You can be a champion)
You can be a master
(You can be a champion)
Don't wait for luck
(You can be a champion)
Dedicate yourself and you can find yourself
(You can be a champion)

Standing in the hall of fame

• -1

ABCD is a trapezium with AB parallel to CD and midpoints of AC and BD be P and  Q

repectively

CONSTRUCTION - Draw a line from D passing through Pand meeting AB in E.

In triangle APE and triangle CPD

angle PAE = angle PCD             ( alternate angles )

Angle PEA = angle PDC              ( alternate angles )

AP = PC                                          ( P is the midpoint of AC )

triangle APE is congruent to triangle CPD

PE = PD                         ( CPCT )

In triangle BED

P is the midpoint of DE             ( proved above )

Q is the midpoint of BD             ( given )

Therefore PQ is parallel to BE                   ( midpoint theorem )

SO PQ is parallel to AB

AB parralel to CD                              ( given )

SO PQ is parallel to CD and AB

HENCE PROVED

• 5
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