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# The above 11 x 11 grid of dots is evenly spaced: each dot is

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Magoosh GMAT Instructor
Joined: 28 Dec 2011
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The above 11 x 11 grid of dots is evenly spaced: each dot is  [#permalink]

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20 May 2013, 11:40
1
8
00:00

Difficulty:

95% (hard)

Question Stats:

37% (02:20) correct 63% (02:12) wrong based on 245 sessions

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Set for yourself a strict 90 second limit for this questions.

The above 11 x 11 grid of dots is evenly spaced: each dot is separated by one unit, vertically or horizontally, from its nearest neighbors. Drawn in the diagram is a single right triangle with legs of length 2 & 1. How many right triangles, congruent to this one, of any orientation, can be formed by three dots from this grid?
(A) 360
(B) 480
(C) 600
(D) 720
(E) 3600

If you are are the type of person that has always felt reasonably comfortable with math, but on GMAT math, you find yourself always running out of time and not able to get to the answer fast enough, then this post is for you:
http://magoosh.com/gmat/2013/how-to-do- ... th-faster/
BTW, it also contains an elegant solution to this problem.

Let me know if anyone has any related questions.

Mike
Attachment:

11 x 11 grid with 2 by 1 triangle.JPG [ 25.35 KiB | Viewed 8423 times ]

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Mike McGarry
Magoosh Test Prep

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Re: The above 11 x 11 grid of dots is evenly spaced: each dot is  [#permalink]

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24 Jun 2013, 12:52
for the grid 9 X 9 we can draw 8 triangles for each point i.e 81*8= 648

for the remaining points it is less than 8 so the maximum triangles we can draw is 81*8 + ( < 40*7)
so the triangle range is between 648 and 918. in the given option only D satisfies this range
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Re: The above 11 x 11 grid of dots is evenly spaced: each dot is  [#permalink]

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24 Jun 2013, 17:15
1
well there are 10 * 9 possible starting places for the triangle.

There are 4 different orientations (IE rotate it 90 degrees four times).

BUt there are twice as many triangles in actuality because we need to invert/flip it.

So 10*9*4*2 = 720
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Re: The above 11 x 11 grid of dots is evenly spaced: each dot is  [#permalink]

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25 Jun 2013, 13:51
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4472
Re: The above 11 x 11 grid of dots is evenly spaced: each dot is  [#permalink]

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25 Jun 2013, 14:29
jasu0072005 wrote:

Please see the entire blog article:
http://magoosh.com/gmat/2013/how-to-do- ... th-faster/
You will find a full explanation there.
Mike
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Mike McGarry
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Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
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Re: The above 11 x 11 grid of dots is evenly spaced: each dot is  [#permalink]

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29 Jun 2013, 05:49
Hi Mike,

Suppose i want to go by another approach,one that does not take rectangles into the picture.

Let us make this our XY coordinate system.
Let us call the triangle ABC so that A has the right angle AB is parallel to the X axis and AC is parallel to the Y axis(in one of the possible orientations).

Case 1: horizontal is 2 units long and vertical is one unit long(for triangle ABC)

Point A: It has 9 options for its X coordinate and 10 options for the Y coordinate. So 90 possible options for A
Point B: will always be restricted by the length given in the question. So only 1 option possible.
Point C:will always be restricted by the vertical length given . so only one option.

So total number of triangles possible in this orientation=90.

SImilarly, we will have 90 triangles possible with this fixed orientation and with the vertical length =2units.

Summing, we have 90+90=180 triangles possible.
Now, this where I get stuck.

Could you help me how to go ahead with the number of different orientations possible?
Maybe with permutations perhaps??

Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4472
Re: The above 11 x 11 grid of dots is evenly spaced: each dot is  [#permalink]

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01 Jul 2013, 11:49
Dear bhang,

You are not counting all orientations. For example, for Case I, 2 units hor & 1 units vert ----- Is B to the right or left of A? Is C above or below A? You see, for any Case I triangle,you are only counting B to the right of A and C above A, but either one of those could still be different.

Mike

12bhang wrote:
Hi Mike,

Suppose i want to go by another approach,one that does not take rectangles into the picture.

Let us make this our XY coordinate system.
Let us call the triangle ABC so that A has the right angle AB is parallel to the X axis and AC is parallel to the Y axis(in one of the possible orientations).

Case 1: horizontal is 2 units long and vertical is one unit long(for triangle ABC)

Point A: It has 9 options for its X coordinate and 10 options for the Y coordinate. So 90 possible options for A
Point B: will always be restricted by the length given in the question. So only 1 option possible.
Point C:will always be restricted by the vertical length given . so only one option.

So total number of triangles possible in this orientation=90.

SImilarly, we will have 90 triangles possible with this fixed orientation and with the vertical length =2units.

Summing, we have 90+90=180 triangles possible.
Now, this where I get stuck.

Could you help me how to go ahead with the number of different orientations possible?
Maybe with permutations perhaps??

_________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
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Re: The above 11 x 11 grid of dots is evenly spaced: each dot is  [#permalink]

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04 Jul 2013, 01:11
mikemcgarry wrote:
Dear bhang,

You are not counting all orientations. For example, for Case I, 2 units hor & 1 units vert ----- Is B to the right or left of A? Is C above or below A? You see, for any Case I triangle,you are only counting B to the right of A and C above A, but either one of those could still be different.

Mike

12bhang wrote:
Hi Mike,

Suppose i want to go by another approach,one that does not take rectangles into the picture.

Let us make this our XY coordinate system.
Let us call the triangle ABC so that A has the right angle AB is parallel to the X axis and AC is parallel to the Y axis(in one of the possible orientations).

Case 1: horizontal is 2 units long and vertical is one unit long(for triangle ABC)

Point A: It has 9 options for its X coordinate and 10 options for the Y coordinate. So 90 possible options for A
Point B: will always be restricted by the length given in the question. So only 1 option possible.
Point C:will always be restricted by the vertical length given . so only one option.

So total number of triangles possible in this orientation=90.

SImilarly, we will have 90 triangles possible with this fixed orientation and with the vertical length =2units.

Summing, we have 90+90=180 triangles possible.
Now, this where I get stuck.

Could you help me how to go ahead with the number of different orientations possible?
Maybe with permutations perhaps??

Hey mike why didnt you use the number of points (121) and used only orientations and inversion?
For eg in case -2<x<4 and 6<y<16, we count all the points?
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Re: The above 11 x 11 grid of dots is evenly spaced: each dot is  [#permalink]

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01 Apr 2015, 07:49
2
2
mikemcgarry wrote:
Set for yourself a strict 90 second limit for this questions.
Attachment:
The attachment 11 x 11 grid with 2 by 1 triangle.JPG is no longer available

The above 11 x 11 grid of dots is evenly spaced: each dot is separated by one unit, vertically or horizontally, from its nearest neighbors. Drawn in the diagram is a single right triangle with legs of length 2 & 1. How many right triangles, congruent to this one, of any orientation, can be formed by three dots from this grid?
(A) 360
(B) 480
(C) 600
(D) 720
(E) 3600

If you are are the type of person that has always felt reasonably comfortable with math, but on GMAT math, you find yourself always running out of time and not able to get to the answer fast enough, then this post is for you:
http://magoosh.com/gmat/2013/how-to-do- ... th-faster/
BTW, it also contains an elegant solution to this problem.

Let me know if anyone has any related questions.

Mike

this can be solved pretty quickly by visualizing the possible orientations of letter 'L' .
as shown in picture, there are 8 possible ways to rotate letter L and for each position we will have 10*9 = 90 such triangles.

so 8*90 = 720 is answer .

Attachments

gmatclub.jpg [ 13.8 KiB | Viewed 5313 times ]

Senior Manager
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The above 11 x 11 grid of dots is evenly spaced: each dot is  [#permalink]

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21 Nov 2016, 10:08
My take on this one.. can be solved in the 90 sec time...
Congruent triangles will fit on each other perfectly essentially triangles of same dimensions.
the only orientations possible are picking 2 spaces i.e.,3 dots horizontally or vertically.
If you see diagonally the length of the sides will be longer. hence only two possible orientations.
Horizontally
out of 10 spaces we need 2 continuous spaces...so 9c2(we need two continuous spaces hence 2 spaces will be considered as 1)
the one space 10c1
Total :9c2 * 10C1 =360
Vertically same :360
total 360*2=720

*Please feel free to point out any errors in my reasoning*
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Re: The above 11 x 11 grid of dots is evenly spaced: each dot is  [#permalink]

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25 Nov 2016, 05:51
@
goforgmat wrote:
My take on this one.. can be solved in the 90 sec time...
Congruent triangles will fit on each other perfectly essentially triangles of same dimensions.
the only orientations possible are picking 2 spaces i.e.,3 dots horizontally or vertically.
If you see diagonally the length of the sides will be longer. hence only two possible orientations.
Horizontally
out of 10 spaces we need 2 continuous spaces...so 9c2(we need two continuous spaces hence 2 spaces will be considered as 1)
the one space 10c1
Total :9c2 * 10C1 =360
Vertically same :360
total 360*2=720

*Please feel free to point out any errors in my reasoning*

Can you please explain why we need 10 spaces horizontally? (or vertically)? can you please walk me through the way of thinking??
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Re: The above 11 x 11 grid of dots is evenly spaced: each dot is  [#permalink]

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25 Nov 2016, 23:14
1
georgekaterji wrote:
@
goforgmat wrote:
My take on this one.. can be solved in the 90 sec time...
Congruent triangles will fit on each other perfectly essentially triangles of same dimensions.
the only orientations possible are picking 2 spaces i.e.,3 dots horizontally or vertically.
If you see diagonally the length of the sides will be longer. hence only two possible orientations.
Horizontally
out of 10 spaces we need 2 continuous spaces...so 9c2(we need two continuous spaces hence 2 spaces will be considered as 1)
the one space 10c1
Total :9c2 * 10C1 =360
Vertically same :360
total 360*2=720

*Please feel free to point out any errors in my reasoning*

Can you please explain why we need 10 spaces horizontally? (or vertically)? can you please walk me through the way of thinking??

We need to connect three dots (horizontally or vertically), three dots continuously has 2 spaces continuously. We cant select any two spaces we need to select two continuous ones so we need to treat 2 continuous as 1 hence selecting 9c2.
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Re: The above 11 x 11 grid of dots is evenly spaced: each dot is  [#permalink]

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22 Jan 2019, 15:02
I got a dumb method but it works.

Forget about points, let's count squares.

Now, add a square beside it, then you are able to draw 4 triangles using two diagonals. (+4)
Then draw another square below our first square, OR beside our second square, we get 8 triangles. (+4)
Then, draw another square that connects to the existing horizontal row or the vertical column. You get 12 triangles (+4)
If you keep this going, you get a vertically inverted L shape.
you can also notice a pattern where with every additional square you get 4 more triangles.

So, according to the grid, to the right of the square 1, we have 9 squares. Below square 1, we also have 9 squares.
A total of 18 additional squares will get you 18*4=72 triangles.

Now that we have the L shaped frame, lets complete the inner part of the grid.
If you draw a square below the second square of the horizontal row OR to the right of the second square of the vertical column, you get 8 more triangles by connecting diagonals with the square above it, or the square beside it.
Then, draw another square beside OR below that, and you would get 8 more triangles.
You'll eventually realize that when a new square is drawn with two of its sides connected to the existing squares, you'd get to create 8 diagonals, instead of 4 from the "L" frame.

So, for the 11x11 grid, we have a total of 100 squares. Subtract the 19 squares we have in the inverted "L" shape, we have 81 squares left.
Since every new square within the inverted "L" frame produces 8 new triangles, we get 81*8=648 triangles.

Total=72+648=720 triangles.

Re: The above 11 x 11 grid of dots is evenly spaced: each dot is   [#permalink] 22 Jan 2019, 15:02
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