prove that the sum of any two sides of a triangle is greater than twice the length of median drawn to the third side


Given: ΔABC in which AD is a median.

To prove: AB + AC > 2AD.

Construction: Produce AD to E, such that AD = DE. Join EC.

Proof: In ΔADB and ΔEDC,

AD = DE (Construction)

BD = BD (D is the mid point of BC)

∠ADB = ∠EDC (Vertically opposite angles)

∴ ΔADB  ΔEDC (SAS congruence criterion)

⇒ AB = ED (CPCT)

In ΔAEC,

AC + ED > AE (Sum of any two sides of a triangles is greater than the third side)

∴ AC + AB > 2AD (AE = AD + DE = AD + AD = 2AD & ED = AB)

Cheers!

 

  • 140

itzz pythogoras theorem

  • -13

pythagoras

  • -16

thanq ...

  • -12

suppose if we take all the angles as x+x+x=180

so x+x=2x which therefore is greater then x

2x>x

easy question.

  • -34


Given: ΔABC in which AD is a median.

To prove: AB + AC > 2AD.

Construction: Produce AD to E, such that AD = DE. Join EC.

Proof: In ΔADB and ΔEDC,

AD = DE (Construction)

BD = BD (D is the mid point of BC)

∠ADB = ∠EDC (Vertically opposite angles)

∴ ΔADB  ΔEDC (SAS congruence criterion)

⇒ AB = ED (CPCT)

In ΔAEC,

AC + ED > AE (Sum of any two sides of a triangles is greater than the third side)

∴ AC + AB > 2AD (AE = AD + DE = AD + AD = 2AD & ED = AB)

Cheers!

 

  • 9

same question appered in my maths paper........

  • -13

 umangbansal600, u were a bit wrong. It is AB=EC, not ED. Thus it is AC+EC>AE. Therfore, AC+AB>2AD

  • 55

 HOW IS IT PYTHOGORAS????????

  • -26

Anshudude is right!

  • -11

 How its a pythogoras????? because in this theorem  Hypotaneouse of a rightangle Triangle is equal to the sum of remaining two sides.

  • -5
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