Everything Maths Grade 10

B C

A

D E

SOLUTION

Step 1: ExtendDEtoFso thatDE=EFand joinF C

B C

A

D

E

F

1

2

Step 2: ProveBCF Dis a parallelogram

In△EADand△ECF:
E^ 1 =E^ 2 (vert opp\s=)
AE =CE (given)
DE =EF (by construction)
)△EAD △ECF (SAS)
)ADE^ =CF E^

But these are alternate interior angles, thereforeBD∥F C

BD =DA (given)

DA =F C (△EAD△ECF)

)BD =F C

)BCF Dis a parallelogram (one pair opp. sides= and∥)

ThereforeDE∥BC.

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

Step 3: Use properties of parallelogramBCF Dto prove thatDE=^12 BC

DF =BC (opp sides∥m)

andDF = 2 (DE) (by construction)

) 2 DE =BC

)DE =^12 BC

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.

Chapter 7. Euclidean geometry 263