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G is the centroid

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G is the centroid

by sanju09 » Mon Feb 22, 2010 5:28 am
In a triangle ABC, AB = AC, and the coordinates of points B and C are (3, 6) and (7, 12), respectively. If G is the centroid of the triangle ABC, which lies on the line x - y = 1, then what are the coordinates of G?
(A) (7, 8)
(B) (8, 7)
(C) (6, 7)
(D) (7, 6)
(E) (-8, -7)
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by ajith » Mon Feb 22, 2010 5:38 am
sanju09 wrote:In a triangle ABC, AB = AC, and the coordinates of points B and C are (3, 6) and (7, 12), respectively. If G is the centroid of the triangle ABC, which lies on the line x - y = 1, then what are the coordinates of G?
(A) (7, 8)
(B) (8, 7)
(C) (6, 7)
(D) (7, 6)
(E) (-8, -7)
Since AB = AC , the angle bisector of angle A will be perpendicular bisector BC

Equation of line BC is x-7/(7-3) = y-12/(12-6)

=> (x-7)/4 = (y-12)/6 => 3x -21 = 2y-24 => 2y-3x-3 =0
Line perpendicular to this will be = 3x + 2y+c and it passes through (5,9)
15+18+c = 0
c =-33

Centroid is the meeting point of line 3x + 2y= 33 and x-y =1
Which is (7,6)
D
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by sanju09 » Mon Feb 22, 2010 5:44 am
ajith wrote:
sanju09 wrote:In a triangle ABC, AB = AC, and the coordinates of points B and C are (3, 6) and (7, 12), respectively. If G is the centroid of the triangle ABC, which lies on the line x - y = 1, then what are the coordinates of G?
(A) (7, 8)
(B) (8, 7)
(C) (6, 7)
(D) (7, 6)
(E) (-8, -7)
Since AB = AC , the angle bisector of angle A will be perpendicular bisector BC

Equation of line BC is x-7/(7-3) = y-12/(12-6)

=> (x-7)/4 = (y-12)/6 => 3x -21 = 2y-24 => 2y-3x-3 =0
Line perpendicular to this will be = 3x + 2y+c and it passes through (5,9)
15+18+c = 0
c =-33

Centroid is the meeting point of line 3x + 2y= 33 and x-y =1
Which is (7,6)
D
[spoiler]IMO B[/spoiler]
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by harsh.champ » Mon Feb 22, 2010 5:45 am
sanju09 wrote:In a triangle ABC, AB = AC, and the coordinates of points B and C are (3, 6) and (7, 12), respectively. If G is the centroid of the triangle ABC, which lies on the line x - y = 1, then what are the coordinates of G?
(A) (7, 8)
(B) (8, 7)
(C) (6, 7)
(D) (7, 6)
(E) (-8, -7)
Centroid is the intersection of the medians.
Actually,the centroid is exactly two-thirds the way along each median.
Put another way, the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex. These lengths are shown on the one of the medians in the figure at the top of the page so you can verify this property for yourself.
Also,AB = AC so,the centroid also lies on the altitude and the perpendicular bisector.
Slope of line joining BC => (12 - 6)/(7-3)
=3/2
Hence,slope of median =-2/3 [m1 x m2 =-1]
Also, it passses thru the pt. (5,9)[Calculated frm avg of tht pts. B and C)
hence, we get the eqn. as 3y = -2x + 37
Putting x-y=1 or x=y+1
we get 5y = 35 or y=7
x=y+1 =8
Hence B is the answer
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