Question

G is centroid of triangle ABC. GE and GF are drawn parallel to AB and AC respectively. Then find A ( GEF ) : A ( ABC )
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Answer 1

A ( GEF ) : A ( ABC ) = 1 : 9

Step-by-step explanation:

Given:

Here GE and GF are drawn parallel to AB and AC, respectively .

Construction: Draw the median AD.

In Δ GEF and Δ ABC,  

∠GEF= ∠ ABC           [Corresponding angles ]

∠GFE= ∠ ACB            [Corresponding angles ]

Therefore, by AA similarity test,  Δ GEF ≅ Δ ABC.

Therefore, $\frac{A( \triangle GEF)}{A( \triangle ABC)} = \frac{GD^2}{AD^2}$      [1]            

[Areas of two similar triangles are in the same proportion as the squares of their medians.]

We know that the centroid divides the median in the ratio 2:1.

Therefore, AG : GD = 2 : 1

⇒AD : GD = 3 : 1

Hence, GD : AD = 1 : 3

⇒$\frac{A(\triangle GEF)}{A(\triangle ABC)} = \frac{1^2}{3^2}$    [from 1]

=$\frac{1}{9}$

A ( ΔGEF) : A ( ΔABC )= 1 : 9

Answer 2

Answer:

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Step-by-step explanation:

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