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Magoosh GMAT Instructor G
Joined: 28 Dec 2011
Posts: 4472
In the grid of dots above, each dot has equal vertical &  [#permalink]

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14 00:00

Difficulty:   75% (hard)

Question Stats: 55% (02:33) correct 45% (02:41) wrong based on 166 sessions

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Attachment: half-square grid of dots with triangle.JPG [ 15.11 KiB | Viewed 6388 times ]

In the grid of dots above, each dot has equal vertical & horizontal spacing from the others. A small 45-45-90 triangle is drawn. Counting this triangle, how many triangles congruent to this one, of any orientation, can be constructed from dots in this grid?
A. 112
B. 120
C. 240
D. 448
E. 480

For a discussion of difficult counting problems, as well as the solution to this problem, see:
http://magoosh.com/gmat/2013/difficult- ... -problems/

Mike _________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
Intern  Joined: 13 Aug 2013
Posts: 43
Re: In the grid of dots above, each dot has equal vertical &  [#permalink]

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Attachment: half-square grid of dots with triangle.JPG [ 19 KiB | Viewed 5673 times ]

Count the number of possible squares (just as the one highlighted above) in figure = 28
Each such square can contain 4 triangles = 28 x 4 = 112.

On the diagonal of the half square we can have another 8 triangles.

So total number of triangles = 112 + 8 = 120
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Re: In the grid of dots above, each dot has equal vertical &  [#permalink]

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mikemcgarry wrote:
Attachment:
half-square grid of dots with triangle.JPG

In the grid of dots above, each dot has equal vertical & horizontal spacing from the others. A small 45-45-90 triangle is drawn. Counting this triangle, how many triangles congruent to this one, of any orientation, can be constructed from dots in this grid?
A. 112
B. 120
C. 240
D. 448
E. 480

For a discussion of difficult counting problems, as well as the solution to this problem, see:
http://magoosh.com/gmat/2013/difficult- ... -problems/

Mike well, sorry that i cant draw any pictures to illustrate but i will try my best to explain with word.
Here is my approach: think of the distance between each dot as 1 unit, i try to image there is a square whose sides are 1 unit. When I try to count all those qualified squares. there are 28 of them. And in each such square, there are 4 ways to place the triangle under the 45-45-90 condition. So there are 28*4=112 of them, plus the left triangles which cannot form a square but also qualify, they are located at the right end from top to bottom, so we get another 8 of those triangles. In total 112+8=120
Board of Directors P
Joined: 17 Jul 2014
Posts: 2509
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30 GPA: 3.92
WE: General Management (Transportation)
In the grid of dots above, each dot has equal vertical &  [#permalink]

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mikemcgarry wrote:
Attachment:
half-square grid of dots with triangle.JPG

In the grid of dots above, each dot has equal vertical & horizontal spacing from the others. A small 45-45-90 triangle is drawn. Counting this triangle, how many triangles congruent to this one, of any orientation, can be constructed from dots in this grid?
A. 112
B. 120
C. 240
D. 448
E. 480

For a discussion of difficult counting problems, as well as the solution to this problem, see:
http://magoosh.com/gmat/2013/difficult- ... -problems/

Mike good question...i almost started to count all possible 45-45-90 triangles that can be drawn..but since we need congruent...the task is easier...
we have 7, 6, 5, 4, 3, 2, 1 - squares that can be created using the dots. in which square, we can have 4 variations of the triangle.
sum of 7 = 7*8/2 = 7*4=28
now, 28*4 = 112. we can eliminate A right away, as we did not count the "outer" possible triangles.
we can draw 8 triangles with the "outer" points.
112+8=120.

B

p.s. is there a way to count all possible variations of 45-45-90 triangle??? I believe it should be possible using the similar method..but would take way too much to solve..
Math Expert V
Joined: 02 Aug 2009
Posts: 7978
Re: In the grid of dots above, each dot has equal vertical &  [#permalink]

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mvictor wrote:
mikemcgarry wrote:
Attachment:
half-square grid of dots with triangle.JPG

In the grid of dots above, each dot has equal vertical & horizontal spacing from the others. A small 45-45-90 triangle is drawn. Counting this triangle, how many triangles congruent to this one, of any orientation, can be constructed from dots in this grid?
A. 112
B. 120
C. 240
D. 448
E. 480

For a discussion of difficult counting problems, as well as the solution to this problem, see:
http://magoosh.com/gmat/2013/difficult- ... -problems/

Mike good question...i almost started to count all possible 45-45-90 triangles that can be drawn..but since we need congruent...the task is easier...
we have 7, 6, 5, 4, 3, 2, 1 - squares that can be created using the dots. in which square, we can have 4 variations of the triangle.
sum of 7 = 7*8/2 = 7*4=28
now, 28*4 = 112. we can eliminate A right away, as we did not count the "outer" possible triangles.
we can draw 8 triangles with the "outer" points.
112+8=120.

B

p.s. is there a way to count all possible variations of 45-45-90 triangle??? I believe it should be possible using the similar method..but would take way too much to solve..

Hi,

ONE way I can think of is -

Look at the Hypotenuse...
the outermost has 9 dots and there on keeps reducing by 1 for each slant line below it..
the outermost cannot make any 45-45-90 triangles..
the second line has 8 dots and each dot can make ONE triangle with the outer most HYP= 8*1..
the Third line has 7 dots and each dot can make TWO triangle with the two hypotenuse outside= 7*2..
similarily 1 dots and that can make 8 triangles with 8 HYP= 1*8...
TOTAL = $$8*1+7*2+6*3+5*4+4*5+3*6+2*7+1*8 = 2 ( 8*1+7*2+6*3+5*4 ) = 2(8+14+18+20) = 2*60 = 120$$
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Joined: 05 Mar 2015
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Re: In the grid of dots above, each dot has equal vertical &  [#permalink]

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mikemcgarry wrote:
Attachment:
The attachment half-square grid of dots with triangle.JPG is no longer available

In the grid of dots above, each dot has equal vertical & horizontal spacing from the others. A small 45-45-90 triangle is drawn. Counting this triangle, how many triangles congruent to this one, of any orientation, can be constructed from dots in this grid?
A. 112
B. 120
C. 240
D. 448
E. 480

For a discussion of difficult counting problems, as well as the solution to this problem, see:
http://magoosh.com/gmat/2013/difficult- ... -problems/

Mike a four dots will contribute to 4 congruent trangles as shown taking each set os dots as a new base
secondly every extreme right side 3 dots will form only 1 triangle
first row will make 1 triangle
second row will make 4+1=5 triangle
third row =9 triangles
a pattern here is every new row will add 4 new triangles.
there are 8 rows
total triangles=(1+5+9+....29)=120
Ans B
Attachments half-square grid of dots with triangle.JPG [ 25.36 KiB | Viewed 5082 times ]

Intern  B
Joined: 09 Dec 2016
Posts: 14
Concentration: Leadership, General Management
Schools: ISB '19 (D)
Re: In the grid of dots above, each dot has equal vertical &  [#permalink]

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Why bigger triangles are not considered in this? Does congruent means equal? I thought it to be similar triangle.
Magoosh GMAT Instructor G
Joined: 28 Dec 2011
Posts: 4472
Re: In the grid of dots above, each dot has equal vertical &  [#permalink]

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ashokjha1986 wrote:
Why bigger triangles are not considered in this? Does congruent means equal? I thought it to be similar triangle.

Dear ashokjha1986,

i'm happy to respond. These are good terms to know, as a GMAT Quant problem might mention them:
1) congruent = same shape, same size (equal angles, equal lengths)
2) similar = same shape, different size (equal angles, proportional lengths)

Does this make sense?
Mike _________________
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: In the grid of dots above, each dot has equal vertical &  [#permalink]

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