G is the centroid of triangleABC. GE and GF are drawn parallel to
AB and AC respectively.
Find A(AGEF): A(AABC).
Answers
Answer:
AABC it is second answer of that
The ratio of the Area of (∆GEF): Area of (∆ABC) is 1/9.
Step-by-step explanation:
It is given that,
GE // AB and GF // AC
G is the centroid of the triangle ABC
Step 1:
Consider, Δ GEF and Δ ABC, we have
∠GEF= ∠ ABC ……… [Corresponding angles, since GE//AB]
∠GFE= ∠ ACB ……. [Corresponding angles, since GF//AC]
∴ By AA similarity, Δ GEF ~ Δ ABC
We know that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.
Here we have AD as the median of Δ ABC and GD as the median of Δ GEF, since point G is given as the centroid of Δ ABC and we know that centroid is the point where the medians of a triangle meet.
∴ [Area(Δ GEF)] / [Area (Δ ABC)] = [GD]² / [AD]² …….. (i)
Step 2:
We also know that a centroid divides the median of a triangle in the ratio of 2:1.
∴ AG : GD = 2 : 1
⇒ GD : AD = 1 : (2+1)
⇒ GD:AD = 1:3 ….. (ii)
From (i) & (ii), we get
[Area(Δ GEF)] / [Area (Δ ABC)] = [1]² / [3]²
⇒ [Area(Δ GEF)] / [Area (Δ ABC)] = 1/9
Thus, the ratio of the Area of (∆GEF): Area of (∆ABC) is 1:9.
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Also view:
THE MEDIANS BE & CF of a triangle ABC intersect at G . prove that area of triangle GBC = area of quadrilateral AFGE .
https://brainly.in/question/83532
Prove that the centroid of a triangle divides the median in the ratio of 2:1 .
https://brainly.in/question/2271338