F.3 MathProblem

[匿名]

2015-03-21 7:42 am

1) In the figure, D,E and F are the mid-points of AB,AC and BC respectively.
(a)Show that DECF is a parallelogram.
(b) Let G be the centroid of triangle ABC. David claims that G is also the centroid of tiangle DEF. Is he correct? Explain your answer.
Figure:http://postimg.org/image/5s9lpol4t/

Need steps, plz!

回答 (1)

用戶10012

2015-03-21 4:40 pm

✔ 最佳答案

1)
(a) E is the mid - point of AC and D is the mid - point of AB, so by mid - point theorem, DE//BC.
Similarly, F is the mid -point of BC, again by mid - point theorem, DF//EC.
Therefore, DECF is a parallelogram (opposite sides //).
(b)
By definition of centroid, G is the intersecting point of the medians of a triangle.
So G is the intersecting point of DC and BE.
Let DC cuts EF at X, FX = DB/2 (mid - point thm.)
EX = AD/2 ( mid - point thm.)
But AD = DB because D is the mid - point of AB, that means FX = EX.
So X is the mid - point of EF, that means DX is one of the medians of triangle DEF or DC is also the median of triangle DEF.
Similarly, BE is also the median of triangle DEF.
In conclusion, G is also the centroid of triangle DEF. David is correct.

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