Prove that medians of a trangle divided each other in the same ratio of 2:1.
Answers
Step-by-step explanation:
Let there be a triangle ABC, with medians AD , BE and CF respectively.
To prove : AO/DO =CO/OF=BO/OE=2/1
Construction: Draw a line from point D to the side AC such that intersect AC at point X and such that DX is parallel to BE.
So, now we start the proof: In triangle BCE, DX is parallel to BE, so CD/BD =CX/XE ( by basic proportionality theorem in which if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio) .
Now BD=CD as AD is a median on side BC. So, CD/CD=CX/XE
1/1=CX/XE
CX=XE (by cross multiplication)
Now in triangle ADC, OE is parallel to DX. So,
AE/EX=AO/DO
Now AE=AC/2 (because BE is a median on side AC)
And, EX=CE/2 ( because we have proved above that CE=EX, so X is the mid point of CE)
Hence, AC/2 /CE/2=AO/DO.
Now, 2 gets cancel so AC/CE=AO/DO
2CE/CE=AO/DO ( as BE is the median on side AC so AC=2CE)
Now, CE gets cancel so,
similarly , we can proof :-
AO/DO=CO/OF=BO/OE=2/1.
Hence, proved
Hope it helps you
Answer:
Step-by-step explanation:
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