Question

what is midpoint theorem

Answer 1

Extend DE to F so that DE=EF and join FC
Prove BCFD is a parallelogram

In △EAD and △ECF:

E‸1AEDE∴△EAD∴AD‸E=E‸2=CE=EF≡△ECF=CF‸E(vert.opp.∠'s)(given)(byconstruction)(SAS)(1)

But these are alternate interior angles, therefore BD∥FC

BDDA∴BD∴BCFDisaparallelogram=DA=FC=FC(onepairopp.sides=and∥)(given)(△EAD≡△ECF)(2)

Therefore DE∥BC.

We conclude that the line joining the two mid-points of two sides of a triangle is parallel to the third side.

Use properties of parallelogram BCFD to prove that DE=12BCDFandDF∴2DE∴DE=BC=2(DE)=BC=12BC(opp.sidesparmequal)(byconstruction)(3)

We conclude that the line joining the mid-point of two sides of a triangle is equal to half the length of the third side.