Question

Triangle A B C has centroid G. Lines are drawn from each point through the centroid to the midpoint of the opposite side to form line segments A E, B F, and C D. The length of line segment A G is 2 x + 10, the length of line segment F G is 2 x minus 1, and the length of line segment G B is 3 x + 6.

G is the centroid of triangle ABC.

What is the length of AE?

units

Answer 1

Answer:

28/3

Step-by-step explanation:

A median of a triangle is a segment from a vertex to the midpoint of the opposite  side. The three medians of a triangle are concurrent. The point of concurrency, called  the centroid, is inside the triangle and The centroid of a triangle is two-thirds of the  distance from each vertex to the midpoint of  the opposite side.

AG = 2x+10

FG = 2x -1

GB= 3x + 6

As we already knew in the principle, so GB= 2FG

<=> 3x + 6  = 2( 2x -1)

<=> 4x= 8

<=> x =2

So, AG = 2*2+10 =14

But AG = 2/3AE, so AE = 2/3*14 = 28/3

Answer 2

Answer:

The length of AE is 39 units.

Step-by-step explanation:

Given:

G is the centroid of a triangle ABC.

Lines are drawn from each point through the centroid to the midpoint of the opposite side to form line segments AE, BF, and CD.

The length of line segment A G is 2 x + 10, the length of line segment F G is 2 x minus 1, and the length of line segment G B is 3 x + 6

To Find: Length of the AE.

The centroid of the triangle is always cut the line in the ratio of 2:1. For clarification, the image is provided.

Hence,

G divides the line BF into a 2:1 ratio.

Thus,

$\begin{aligned}&2FG=BG\\&2(2x-1)=3x+6\\&4x-2=3x+6\\&x=8 \end{aligned}$

Also,

G divides the line AE into a 2:1 ratio.

Therefore,

$AG=2GE\\GE=AG/2$

Line segment AE is the sum of AG and GE.

Thus,

$AE=AG+GE\\AE=AG+AG/2\\AE=3AG/2\\AE=3(2x+10)/2\\AE=39$

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