In the grid of dots above, each dot has equal vertical &

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Mike@Magoosh: GMAT Instructor; Posts: 768; Joined: Wed Dec 28, 2011 4:18 pm; Location: Berkeley, CA; Thanked: 387 times; Followed by:140 members

In the grid of dots above, each dot has equal vertical &

by Mike@Magoosh » Wed Nov 20, 2013 4:07 pm

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 complete solution to this problem, as well as discussion of hard counting problems in general, see:
https://magoosh.com/gmat/2013/difficult- ... -problems/

Mike

Magoosh GMAT Instructor
https://gmat.magoosh.com/

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Source: Beat The GMAT — Problem Solving | Wed Nov 20, 2013 4:07 pm

Mathsbuddy: Master | Next Rank: 500 Posts; Posts: 447; Joined: Fri Nov 08, 2013 7:25 am; Thanked: 25 times; Followed by:1 members

by Mathsbuddy » Wed Nov 20, 2013 11:49 pm

There are 7+6+....+1 unit squares = 7/2 * (7+1) = 28 using Sum = n/2 * (n + 1)
Within each square there are four different possible (rotated) orientations of the congruent triangle.
Also, there are 8 more congruent triangles along the hypotenuse.

Total = 28 * 4 + 8 = 120

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vipulgoyal: Master | Next Rank: 500 Posts; Posts: 468; Joined: Mon Jul 25, 2011 10:20 pm; Thanked: 29 times; Followed by:4 members

by vipulgoyal » Thu Nov 21, 2013 11:15 pm

shouldnt we need to consider additional trangles in shaded squres in the attached file,
my take

120 + {4+8+12+16+20(trangles formed within those squres formed by joining three verticle and three horiz dots)+6(from right side trangles) + 4+8+12 (trangles formed within those squres formed by joining four verticle and four horiz dots)+3(from right side trangles)+ 4 (trangles formed within those squres formed by joining five verticle and five horiz dots)+1(from right side trangles)

[spoiler] 120 + 78 = 198 not in option [/spoiler]

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Mathsbuddy: Master | Next Rank: 500 Posts; Posts: 447; Joined: Fri Nov 08, 2013 7:25 am; Thanked: 25 times; Followed by:1 members

by Mathsbuddy » Fri Nov 22, 2013 12:01 am

vipulgoyal wrote:shouldnt we need to consider additional trangles in shaded squres in the attached file,
my take

120 + {4+8+12+16+20(trangles formed within those squres formed by joining three verticle and three horiz dots)+6(from right side trangles) + 4+8+12 (trangles formed within those squres formed by joining four verticle and four horiz dots)+3(from right side trangles)+ 4 (trangles formed within those squres formed by joining five verticle and five horiz dots)+1(from right side trangles)

[spoiler] 120 + 78 = 198 not in option [/spoiler]

Hi there,

That's what I thought when I first saw the question, but the key adjective is "CONGRUENT" which basically means "exactly the same shape and size". If the question had said "SIMILAR" triangles then all the 45-45-90 triangles would need to be considered like you say. If it just said triangles (i.e. not necessarily congruent/similar) then there would be plenty more triangular choices.

I hope this helps

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theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pm; Thanked: 448 times; Followed by:34 members; GMAT Score:650

by theCodeToGMAT » Fri Nov 22, 2013 12:11 am

vipulgoyal wrote:shouldnt we need to consider additional trangles in shaded squres in the attached file,
my take

120 + {4+8+12+16+20(trangles formed within those squres formed by joining three verticle and three horiz dots)+6(from right side trangles) + 4+8+12 (trangles formed within those squres formed by joining four verticle and four horiz dots)+3(from right side trangles)+ 4 (trangles formed within those squres formed by joining five verticle and five horiz dots)+1(from right side trangles)

[spoiler] 120 + 78 = 198 not in option [/spoiler]

According to me, if the question had said "Similar"

Total single cube based = (7)(8)/2 = 28 ==> 28*4 + 8 = 120

Total 4 cube based square = (5)(6)/2 = 15 ==> 15*4 + 7 = 67

total 9 cube based square = (3)(4)/2 = 6 ==> 6*4 + 6 = 30

Total 16 cube based square = 1 cube ==> 1*4 + 5 = 9

Total = 226

Last edited by theCodeToGMAT on Fri Nov 22, 2013 2:43 am, edited 2 times in total.

R A H U L

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Mathsbuddy: Master | Next Rank: 500 Posts; Posts: 447; Joined: Fri Nov 08, 2013 7:25 am; Thanked: 25 times; Followed by:1 members

by Mathsbuddy » Fri Nov 22, 2013 12:17 am

theCodeToGMAT wrote:
vipulgoyal wrote:shouldnt we need to consider additional trangles in shaded squres in the attached file,
my take

120 + {4+8+12+16+20(trangles formed within those squres formed by joining three verticle and three horiz dots)+6(from right side trangles) + 4+8+12 (trangles formed within those squres formed by joining four verticle and four horiz dots)+3(from right side trangles)+ 4 (trangles formed within those squres formed by joining five verticle and five horiz dots)+1(from right side trangles)

[spoiler] 120 + 78 = 198 not in option [/spoiler]
We need congruent triangle with 45-45-90 measurement.. if you are trying to make those triangle in the shaded region then you are again considering those triangles.

If we scan each point on the left most column and consider possible 45-45-90 triangles, then:

(8)(9)/2 = 36
(7)(8)/2 = 28
(6)(7)/2 = 21
(5)(6)/2 = 15
(4)(5)/2 = 10
(3)(4)/2 = 6
(2)(3)/2 = 3
(1)(2)/2 = 1
= 120....

I'm not sure I know what you mean.
Congruent triangles must be the SAME SIZE.

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sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 am; Location: India; Thanked: 267 times; Followed by:80 members; GMAT Score:760

by sanju09 » Fri Nov 22, 2013 12:18 am

Mike@Magoosh wrote:
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 complete solution to this problem, as well as discussion of hard counting problems in general, see:
https://magoosh.com/gmat/2013/difficult- ... -problems/

Mike

Nice question Mike! I agree with Mathsbuddy's approach and answer. It saves time that's the essence.

The mind is everything. What you think you become. -Lord Buddha

Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001

www.manyagroup.com

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theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pm; Thanked: 448 times; Followed by:34 members; GMAT Score:650

by theCodeToGMAT » Fri Nov 22, 2013 12:20 am

Mathsbuddy wrote:
theCodeToGMAT wrote:
vipulgoyal wrote:shouldnt we need to consider additional trangles in shaded squres in the attached file,
my take

120 + {4+8+12+16+20(trangles formed within those squres formed by joining three verticle and three horiz dots)+6(from right side trangles) + 4+8+12 (trangles formed within those squres formed by joining four verticle and four horiz dots)+3(from right side trangles)+ 4 (trangles formed within those squres formed by joining five verticle and five horiz dots)+1(from right side trangles)

[spoiler] 120 + 78 = 198 not in option [/spoiler]
We need congruent triangle with 45-45-90 measurement.. if you are trying to make those triangle in the shaded region then you are again considering those triangles.

If we scan each point on the left most column and consider possible 45-45-90 triangles, then:

(8)(9)/2 = 36
(7)(8)/2 = 28
(6)(7)/2 = 21
(5)(6)/2 = 15
(4)(5)/2 = 10
(3)(4)/2 = 6
(2)(3)/2 = 3
(1)(2)/2 = 1
= 120....
I'm not sure I know what you mean.
Congruent triangles must be the SAME SIZE.

Yep, i was just restating what's present in the question; Mike has stated that drawn triangle is "45-45-90"

R A H U L

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sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 am; Location: India; Thanked: 267 times; Followed by:80 members; GMAT Score:760

by sanju09 » Fri Nov 22, 2013 12:22 am

theCodeToGMAT wrote:
vipulgoyal wrote:shouldnt we need to consider additional trangles in shaded squres in the attached file,
my take

120 + {4+8+12+16+20(trangles formed within those squres formed by joining three verticle and three horiz dots)+6(from right side trangles) + 4+8+12 (trangles formed within those squres formed by joining four verticle and four horiz dots)+3(from right side trangles)+ 4 (trangles formed within those squres formed by joining five verticle and five horiz dots)+1(from right side trangles)

[spoiler] 120 + 78 = 198 not in option [/spoiler]
We need congruent triangle with 45-45-90 measurement.. if you are trying to make those triangle in the shaded region then you are again considering those triangles.

If we scan each point on the left most column and consider possible 45-45-90 triangles, then:

(8)(9)/2 = 36
(7)(8)/2 = 28
(6)(7)/2 = 21
(5)(6)/2 = 15
(4)(5)/2 = 10
(3)(4)/2 = 6
(2)(3)/2 = 3
(1)(2)/2 = 1
= 120....

We need congruent triangles of the size mentioned in the dots. But, you're lucky on this occasion, Rahul. Read this in the stem, 'how many triangles congruent to this one'.

The mind is everything. What you think you become. -Lord Buddha

Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001

www.manyagroup.com

Quote

Mathsbuddy: Master | Next Rank: 500 Posts; Posts: 447; Joined: Fri Nov 08, 2013 7:25 am; Thanked: 25 times; Followed by:1 members

by Mathsbuddy » Fri Nov 22, 2013 12:25 am

theCodeToGMAT wrote:
Mathsbuddy wrote:
theCodeToGMAT wrote:
vipulgoyal wrote:shouldnt we need to consider additional trangles in shaded squres in the attached file,
my take

120 + {4+8+12+16+20(trangles formed within those squres formed by joining three verticle and three horiz dots)+6(from right side trangles) + 4+8+12 (trangles formed within those squres formed by joining four verticle and four horiz dots)+3(from right side trangles)+ 4 (trangles formed within those squres formed by joining five verticle and five horiz dots)+1(from right side trangles)

[spoiler] 120 + 78 = 198 not in option [/spoiler]
We need congruent triangle with 45-45-90 measurement.. if you are trying to make those triangle in the shaded region then you are again considering those triangles.

If we scan each point on the left most column and consider possible 45-45-90 triangles, then:

(8)(9)/2 = 36
(7)(8)/2 = 28
(6)(7)/2 = 21
(5)(6)/2 = 15
(4)(5)/2 = 10
(3)(4)/2 = 6
(2)(3)/2 = 3
(1)(2)/2 = 1
= 120....
I'm not sure I know what you mean.
Congruent triangles must be the SAME SIZE.
Yep, i was just restating what's present in the question; Mike has stated that drawn triangle is "45-45-90"

I see. Yes similar or congruent shapes must have the same angles. Congruent shapes must have the same length of side too.

See https://www.math.com/school/subject3/les ... 3L1GL.html

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theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pm; Thanked: 448 times; Followed by:34 members; GMAT Score:650

by theCodeToGMAT » Fri Nov 22, 2013 12:25 am

sanju09 wrote: We need congruent triangles of the size mentioned in the dots. But, you're lucky on this occasion, Rahul. Read this in the stem, 'how many triangles congruent to this one'.

Oh yes, I mixed congruent with Similar Triangle.....

R A H U L

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Mathsbuddy: Master | Next Rank: 500 Posts; Posts: 447; Joined: Fri Nov 08, 2013 7:25 am; Thanked: 25 times; Followed by:1 members

by Mathsbuddy » Fri Nov 22, 2013 12:31 am

Ignore this message!

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vipulgoyal: Master | Next Rank: 500 Posts; Posts: 468; Joined: Mon Jul 25, 2011 10:20 pm; Thanked: 29 times; Followed by:4 members

by vipulgoyal » Fri Nov 22, 2013 1:23 am

if it would be similar, it would be more intresting

Quote

theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pm; Thanked: 448 times; Followed by:34 members; GMAT Score:650

by theCodeToGMAT » Fri Nov 22, 2013 2:45 am

vipulgoyal wrote:if it would be similar, it would be more intresting

R A H U L

Quote

Mathsbuddy: Master | Next Rank: 500 Posts; Posts: 447; Joined: Fri Nov 08, 2013 7:25 am; Thanked: 25 times; Followed by:1 members

by Mathsbuddy » Fri Nov 22, 2013 7:26 am

theCodeToGMAT wrote:
vipulgoyal wrote:if it would be similar, it would be more intresting
According to me, if the question had said "Similar"

Total single cube based = (7)(8)/2 = 28 ==> 28*4 + 8 = 120

Total 4 cube based square = (5)(6)/2 = 15 ==> 15*4 + 7 = 67

total 9 cube based square = (3)(4)/2 = 6 ==> 6*4 + 6 = 30

Total 16 cube based square = 1 cube ==> 1*4 + 5 = 9

Total = 226

Indeed it would be much more interesting, but too complicated for GMat as there are more than 226 similar triangles available! In the attached diagram I have shown a pink alternative similar 45-45-90 triangle in various orientations and 2 sizes. The green triangle introduces the fact that even more orientations of 45-45-90 triangle are possible. There's a challenge for you!

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