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For any triangle T in the xy–coordinate plan, the center of T is defin

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For any triangle T in the xy–coordinate plan, the center of T is defin  [#permalink]

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New post 29 Oct 2014, 08:25
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Tough and Tricky questions: Coordinate Geometry.



For any triangle T in the xy–coordinate plan, the center of T is defined to be the point whose x–coordinate is the average (arithmetic mean) of the x–coordinates of the vertices of T and whose y–coordinate is the average of the y–coordinates of the vertices of T. If a certain triangle has vertices at the points (0,0) and (6,0) and center at the point (3,2), what are the coordinates of the remaining vertex?

A. (3,4)
B. (3,6)
C. (4,9)
D. (6,4)
E. (9,6)

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Re: For any triangle T in the xy–coordinate plan, the center of T is defin  [#permalink]

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New post 04 Nov 2014, 09:34
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Hi Bunuel,

Great question. The best mode of attack is to Dive In.

Let's start by labeling the three vertices of triangle T as a, b, and c. We know pt a is at (0,0), b is at (6,0), and c is unknown, (cx, cy).

The center, as defined in the problem, is the arithmetic mean of the x and y coordinates individually. Let's write that out as a formula, where the center of triangle T is labeled as pt m at (mx, my).

mx = (ax + bx + cx)/3
my = (ay + by + cy)/3

We know from the problem that pt m is at (3,2), so let's plug in first for the x-coordinate, cx:
3 = (0 + 6 + cx)/3
With some arithmetic, we can solve this to see that cx = 3.

Now, let's do the same for the y-coordinate, cy:
2 = (0 + 0 + cy)/3
So... cy = 6.

Putting these together, we now have that the coordinate of the missing vertex is at (cx, cy), or (3,6)... Answer Choice B.

Hope this helps!
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Re: For any triangle T in the xy–coordinate plan, the center of T is defin  [#permalink]

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New post 09 Nov 2014, 13:57
Bunuel wrote:

Tough and Tricky questions: Coordinate Geometry.



For any triangle T in the xy–coordinate plan, the center of T is defined to be the point whose x–coordinate is the average (arithmetic mean) of the x–coordinates of the vertices of T and whose y–coordinate is the average of the y–coordinates of the vertices of T. If a certain triangle has vertices at the points (0,0) and (6,0) and center at the point (3,2), what are the coordinates of the remaining vertex?

A. (3,4)
B. (3,6)
C. (4,9)
D. (6,4)
E. (9,6)



Notice that (3,2) lies on one the median which divides the side connecting [0,0] & [6,0]....and it is give [3,2] is the centre...we know that centriod will divide the median in the ratio 2:1 .....the smaller part of the ratio is 2 here, so the bigger must be 4....and hence the point is [3,6]. Now add the x of all vertices and y of all vertices , find the averages and cross check.

Thus answer B [3,6].


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Re: For any triangle T in the xy–coordinate plan, the center of T is defin  [#permalink]

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New post 27 Apr 2015, 07:42
Can I solve this way?

let the vertices be A, B, and C
A=(0,0), B=(6,0), and since the central is (3,2) then C=(3,n)

sum of x=0+6+3=9; avg=9/3=3
sum of y=0+0+n=; avg=n/3=2 (since the center is (3,2))
thus n=3*2=6

So the answer is (3,6) B
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Re: For any triangle T in the xy–coordinate plan, the center of T is defin  [#permalink]

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New post 14 Jul 2016, 11:12
Bunuel wrote:

Tough and Tricky questions: Coordinate Geometry.



For any triangle T in the xy–coordinate plan, the center of T is defined to be the point whose x–coordinate is the average (arithmetic mean) of the x–coordinates of the vertices of T and whose y–coordinate is the average of the y–coordinates of the vertices of T. If a certain triangle has vertices at the points (0,0) and (6,0) and center at the point (3,2), what are the coordinates of the remaining vertex?

A. (3,4)
B. (3,6)
C. (4,9)
D. (6,4)
E. (9,6)


Seems like a difficult question , but actually it can be solved in less than 30 seconds.
First plot all the given (X,Y) pair.
You will get the base of the triangle with length of 6 and mid point of triangle at 3,3
Now the vortex will be double at the Y coordinate of the midpoint
Double of 3= 6 (y=3)
Average of X is already 3
Therefore the X,Y pair will become 3,6
Answer is B
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Re: For any triangle T in the xy–coordinate plan, the center of T is defin  [#permalink]

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New post 15 Jul 2016, 14:10
LogicGuru1 wrote:
Bunuel wrote:

Tough and Tricky questions: Coordinate Geometry.



For any triangle T in the xy–coordinate plan, the center of T is defined to be the point whose x–coordinate is the average (arithmetic mean) of the x–coordinates of the vertices of T and whose y–coordinate is the average of the y–coordinates of the vertices of T. If a certain triangle has vertices at the points (0,0) and (6,0) and center at the point (3,2), what are the coordinates of the remaining vertex?

A. (3,4)
B. (3,6)
C. (4,9)
D. (6,4)
E. (9,6)


Seems like a difficult question , but actually it can be solved in less than 30 seconds.
First plot all the given (X,Y) pair.
You will get the base of the triangle with length of 6 and mid point of triangle at 3,3
Now the vortex will be double at the Y coordinate of the midpoint
Double of 3= 6 (y=3)
Average of X is already 3
Therefore the X,Y pair will become 3,6
Answer is B
Image



The center is 3,2.

Actually the average is of 3 data points, that is why it is 6 because (0+0+y)/3=2 y=6.

X remains at 3 because
(0 + 6 + x) /3 = 3
6+x=9
x=3

At least it made sense to me this way.
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Re: For any triangle T in the xy–coordinate plan, the center of T is defin  [#permalink]

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New post 15 Jul 2016, 21:58
Ilomelin wrote:
LogicGuru1 wrote:
Bunuel wrote:

Tough and Tricky questions: Coordinate Geometry.



For any triangle T in the xy–coordinate plan, the center of T is defined to be the point whose x–coordinate is the average (arithmetic mean) of the x–coordinates of the vertices of T and whose y–coordinate is the average of the y–coordinates of the vertices of T. If a certain triangle has vertices at the points (0,0) and (6,0) and center at the point (3,2), what are the coordinates of the remaining vertex?

A. (3,4)
B. (3,6)
C. (4,9)
D. (6,4)
E. (9,6)


Seems like a difficult question , but actually it can be solved in less than 30 seconds.
First plot all the given (X,Y) pair.
You will get the base of the triangle with length of 6 and mid point of triangle at 3,3
Now the vortex will be double at the Y coordinate of the midpoint
Double of 3= 6 (y=3)
Average of X is already 3
Therefore the X,Y pair will become 3,6
Answer is B
Image



The center is 3,2.

Actually the average is of 3 data points, that is why it is 6 because (0+0+y)/3=2 y=6.

X remains at 3 because
(0 + 6 + x) /3 = 3
6+x=9
x=3

At least it made sense to me this way.


Yup , you are right

Average should should be 0+3+6=9/3
= 3
Double of 3=6
so vortex will be (x,y)=(3,6)
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FINAL GOODBYE :- 17th SEPTEMBER 2016. .. 16 March 2017 - I am back but for all purposes please consider me semi-retired.

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Re: For any triangle T in the xy–coordinate plan, the center of T is defin  [#permalink]

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New post 22 Feb 2017, 18:06
I think its a very simple averages question. Median point (3,2) should be equal to {(x1+x2+x3)/3, (y1+y2+y3)/3}.
We have (x1,y1) and (x2,y2) as (0,0) and (6,0).
Thus, 3= (0+6+x3)/3 => x3 = 3
& 2= (0+0+y3)/3 => y3 = 6.
The point, therefore, is (3,6). Option B.
Re: For any triangle T in the xy–coordinate plan, the center of T is defin &nbs [#permalink] 22 Feb 2017, 18:06
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