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G is the centroid (the centroid of a triangle is the point of intersec

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Bunuel
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G is the centroid (the centroid of a triangle is the point of intersec [#permalink] New post   11 Jun 2020, 03:22
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G is the centroid (the centroid of a triangle is the point of intersection of its medians) of △ABC. GE and GF are drawn parallel to AB and AC respectively. What is the ratio of the area of (△ GEF) to the area of (△ABC)?


A. 1:3

B. 1:4

C. 1:9

D. 1:12

E. 1:16


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Re: G is the centroid (the centroid of a triangle is the point of intersec [#permalink] New post   11 Jun 2020, 04:06
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Bunuel wrote:

G is the centroid (the centroid of a triangle is the point of intersection of its medians) of △ABC. GE and GF are drawn parallel to AB and AC respectively. What is the ratio of the area of (△ GEF) to the area of (△ABC)?


A. 1:3

B. 1:4

C. 1:9

D. 1:12

E. 1:16[/url]


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Join AG and further to D on BC. Now Ad is the median and the point G will divide it into ratio 2:1
Quote:
Property- The median is a line that joins the midpoint of a side and the opposite vertex of the triangle. The centroid of the triangle separates the median in the ratio of 2: 1.

SO AG=2x and GD = x...
Let us compare the area of ABD and EGD...
Quote:
Since the two triangles are similar, EG||AB, so teh area of two triangles will be in the ratio of square of two similar sides

So AD^2:GD^2=(2x+x)^2:x^2=9:1

Similarly for ACD and CGD..

SO the ratio of the area of (△ GEF) to the area of (△ABC) = 1:9

C
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Re: G is the centroid (the centroid of a triangle is the point of intersec [#permalink] New post   11 Jun 2020, 05:15
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The centroid of a triangle divides the median in the ratio of 2:1 from the vertex to the opposite side. -----(1)
Since all 3 sides of GEF are || to corresponding sides of ABC, both the triangles are similar.
Extend AG to meet BC at D. From equation (1), AG:GD = 2:1 => AG:AD = 1:3

Similar triangles are similar, height and base of the triangles must be in the same ratio as that of the median. So, the ratios of height and base are 1:3.

Area of GEF : 0.5*b*h
Area of ABC : 0.5*3b*3h = 9*Area of GEF

Thus, ratio of their areas is 1:9

Ans: C
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Re: G is the centroid (the centroid of a triangle is the point of intersec [#permalink] New post   11 Jun 2020, 11:09
Area(Tri. GEF) : Area ( Tri. ABC) = ?
Draw median from A to middle of EF i.e E"

The median is a line that joins the midpoint of a side and the opposite vertex of the triangle. The centroid of the triangle separates the median in the ratio of 2: 1.
E"G / E"A = 1/3

Since the two triangles are similar, EG||AB, so the area of two triangles will be in the ratio of square of two similar sides
Tri E"BA similar to E"EG
E"G/E"A = 1/3
Then Area [Tri. GEF ] / Area [ Tri. ABC] = E"G ^2 / E"A ^2 = 1^2 / 3^2 = 1 / 9

Answer : C
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Re: G is the centroid (the centroid of a triangle is the point of intersec [#permalink] New post   12 Jun 2020, 00:03
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G is the centroid (the centroid of a triangle is the point of intersection of its medians) of △ABC. GE and GF are drawn parallel to AB and AC respectively. What is the ratio of the area of (△ GEF) to the area of (△ABC)?

We know that centroid divide median in 2:1 ratio . Let median AX is divided by G in 2:1 .

So AG:GX=2:1.

now GE and GF are parallel to AB and AC .
so tringle GEX is similar to ABX
and GFX is similar to ACX .

so ratio GEX/ABX = GX^2/AX^2= 1/ 3^2= 1:9

ratio GFX /ACX = GX^2/AX^2= 1/ 3^2= 1:9

Since GFX+GEX = GEF and ABX+ ACX = ABC

so area of GEF / area ofABC = 2/18 = 1:9

so C is the ans ..
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G is the centroid (the centroid of a triangle is the point of intersec [#permalink] New post   12 Jun 2020, 00:26
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Bunuel wrote:

G is the centroid (the centroid of a triangle is the point of intersection of its medians) of △ABC. GE and GF are drawn parallel to AB and AC respectively. What is the ratio of the area of (△ GEF) to the area of (△ABC)?


A. 1:3

B. 1:4

C. 1:9

D. 1:12

E. 1:16


OBSERVATION 1: The question doesn't specify anything about the triangle ABC i.e. ∆ABC may be equilateral as well so for ease of calculation assume that ∆ABC is an equilateral triangle


CONCEPT: in an Equilateral Triangle the Centroid, Incenter, Circumcenter are all at same point and the point divides the height of triangle in ratio 2:1


i.e. \(AG = \frac{2}{3}* Height = (\frac{2}{3})*(\frac{√3}{4})*Side^2\)



i.e. Height of EGF = \(\frac{1}{3}* Height = (\frac{1}{3})*(\frac{√3}{4})*Side^2\)

i.e. Ratio of Heights of similar ∆ABC and ∆EGF = 3:1

i.e.
CONCEPT: Ratio of Area of two similar triangles \(\frac{A_1}{A_2} = (\frac{Side_1}{Side_2})^2\)


i.e. \(\frac{A_1}{A_2} = (\frac{3}{1})^2 = \frac{9}{1}\)

Answer: Option C
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Re: G is the centroid (the centroid of a triangle is the point of intersec [#permalink] New post   13 Jun 2020, 23:07
Kindly see the attachment.
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Re: G is the centroid (the centroid of a triangle is the point of intersec [#permalink] New post   15 Jun 2020, 13:14
EF will be the base with height of H
BC will be the base with height of 3H (as centrois cuts the line in 1:2 ratio).
EF/BC=H/3H (ratio of similar sides).
so EF=1/3BC

the ratio (i'll ignore divide by 2)
1/3H*BC/ 3H*BC=1/9
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Re: G is the centroid (the centroid of a triangle is the point of intersec [#permalink]
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