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# Arrow AB which is a line segment exactly 5 units along with

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Re: Arrow AB which is a line segment exactly 5 units along with [#permalink]
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pallaviisinha wrote:
VeritasPrepKarishma wrote:
harikris wrote:
Arrow AB which is a line segment exactly 5 units along with an arrowhead at A is to be constructed in the xy-plane. The x and y coordinates of A and B are to be integers that satisfy the inequalities 0 ≤ x ≤ 9 and 0 ≤ y ≤ 9. How many different arrows with these properties can be constructed ?

A. 50
B. 168
C. 200
D. 368
E. 536

Consider the diagram.
The arrows could be vertical, horizontal or diagonal.

Attachment:
Ques4.jpg

The vertical arrows are shown by the blue arrows. 5 of them will start from x = 0, 5 from x = 1 and so on till x = 9. So you have 50 of these blue arrows. You have another 50 vertical arrows which are the same arrows but with the arrow head on the opposite end (shown by the red arrow). So you have a total of 100 vertical arrows.
Similarly, you have 100 horizontal arrows.

Now check out the diagonal arrows. One co-ordinate should be of length 3 and another of 4 (so that the arrow length is 5 and all points are integers). Look at the purple arrows. The x co-ordinate is 3 and the y co-ordinate is 4. You can make 7*6 = 42 such arrows. Similarly, you can make 42 arrows with x cor-ordinate as 4 and y co-ordinate as 3. So you have 84 arrows. But you get another set of 84 arrows by keeping the arrows the same but putting the arrow head on the opposite end so you get a total of 2*84 = 168 arrows.

Similarly, you can make arrows in the opposite direction shown by the green arrows. So you have another 168 arrows.

Total = 100 + 100 + 168 + 168 = 536

Hi,

Tried viewing the diagram but it seems diagram cannot be viewed. Please re-upload it, if possible.

Regards,
Pallavi

Diagram is visible in the original post: arrow-ab-which-is-a-line-segment-exactly-5-units-along-with-139558.html#p1127334

Here it is again:

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Arrow AB which is a line segment exactly 5 units along with [#permalink]
"The x co-ordinate is 3 and the y co-ordinate is 4. You can make 7*6 = 42 such arrows. "

Hi Karishma,
Will you please tell me how you got this ?
" You can make 7*6 = 42 such arrows.[/color]"

How you got 7*6 ?
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Re: Arrow AB which is a line segment exactly 5 units along with [#permalink]
sayan640 wrote:
"The x co-ordinate is 3 and the y co-ordinate is 4. You can make 7*6 = 42 such arrows. "

Hi Karishma,
Will you please tell me how you got this ?
" You can make 7*6 = 42 such arrows.[/color]"

How you got 7*6 ?

x is between 0 and 9 (inclusive) and y is between 0 and 9 (inclusive).

So if we need the length of x to be 3, we can start it anywhere from x = 0 to x = 6 (7 cases).
If we start the arrow at x = 7, with a length of 3, it will mean that the end point of the arrow will have x co-ordinate of 10 (which is not allowed).
So for the x co-ordinate, you have 7 options.

Similarly, the length of y should be 4 so we can start it anywhere from y = 0 to y = 5 (6 cases)
If we start the arrow at y = 6, with a length of 4, it will mean that the end point of the arrow will have y co-ordinate of 10 (which is not allowed).
So for the y co-ordinate, you have 6 options.

Hence you get a total of 7 * 6 = 42 acceptable cases.
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Re: Arrow AB which is a line segment exactly 5 units along with [#permalink]
harikris wrote:
Arrow AB which is a line segment exactly 5 units along with an arrowhead at A is to be constructed in the xy-plane. The x and y coordinates of A and B are to be integers that satisfy the inequalities 0 ≤ x ≤ 9 and 0 ≤ y ≤ 9. How many different arrows with these properties can be constructed ?

A. 50
B. 168
C. 200
D. 368
E. 536

Total = 100 (Horizontal Arrows)+ 100 (Vertical Arrows) + 168 (Diagonal Arrows)+ 168 (Diagonal Arrows) = 536
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Arrow AB which is a line segment exactly 5 units along with [#permalink]
Arrow AB which is a line segment exactly 5 units along with an arrowhead at A is to be constructed in the xy-plane. The x and y coordinates of A and B are to be integers that satisfy the inequalities 0 ≤ x ≤ 9 and 0 ≤ y ≤ 9. How many different arrows with these properties can be constructed ?

A. 50
B. 168
C. 200
D. 368
E. 536

root ((x1-x2)^2+(y1-y2)^2) = 25

1 > if |x1-x2| = 5 & |y1-y2| =0
x1 =0 x2 =5
x1 =1 x2 =6
......

x1 =4 x2 =9
x1 =5 X2 =0
x1 =9 x2 = 4
................
so total 10 possibility

for (y1-y2) =0

y1 =0 y2=0
..
y1=9 y2 =9
so total 10 possibility

total 100 combination
2 > if |x1-x2| = 0 & |y1-y2| = 5
again 100 combination

3> if |x1-x2| = 3 & |y1-y2| =4

x1=0 x2=3
........
x1=3 x2= 6/0
x1=4 x2= 7/1
x1=5 x2= 2/8
x1=6 x2 = 9/3
x1 =7 x2= 4
x1 =8 x2= 5
x1 =9 x2= 6

total 14 possibility

(y1-y2) =4
y1=0 y2=4
........
y1=4 y2=0

y1=5 y2 =9
total 12 possibility
combination =14*12 =168 possibility

3> if |x1-x2| = 4 & |y1-y2| =3
combination =14*12 =168 possibility
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Re: Arrow AB which is a line segment exactly 5 units along with [#permalink]
I dont think that questions such as this often come in exams. Even for a 700+ level, this is a rare question.
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Re: Arrow AB which is a line segment exactly 5 units along with [#permalink]
Top Contributor
I think there is a typo? Highlighted in red
monsama wrote:
If A and B have the same x-coordinate then we have 10 pairs of y-coordinate of A and B per x-coordinate. (eg: 1-5, 2-6...) => 10*10 = 100 arrows.
Similarly, if A and B have the same y-coordinate then we have another 100 arrows.
If A(a,c) and B(b,d) don't have the same x-coordinate or y-coordinate then either |a-b|=3,|c-d|=4 or |a-b|=4,|c-d|=3
In the first case, there are 14 pairs of x-coordinate, and 12 pairs of y-coordinate. => 14*12 = 168 arrows.
Similarly in the second case, there are 168 arrows.
Therefore, We have 100 + 100 + 168 + 168 = 536 arrows.
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Re: Arrow AB which is a line segment exactly 5 units along with [#permalink]
daviesj wrote:
I didn't get the explaination....
what i did was i formed the grid on xy plane with info provided. total grid points i got were 100 and we need to select 2 points to form an arrow...so 100C2 : 4950...which is nowhere near the answer....whr exactly i m making the mistake?

Posted from my mobile device

What You did not consider is length should be 5 units.
The way by which You are selecting could lead to any length. So select a way of making length of line equal to 5 units.
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Re: Arrow AB which is a line segment exactly 5 units along with [#permalink]
SoumiyaGoutham wrote:
Now check out the diagonal arrows. One co-ordinate should be of length 3 and another of 4 (so that the arrow length is 5 and all points are integers),..

I hope that u understood the non diagonal type of arrows. Foe example A(1,0) and B(6,0). So the length will be 5 units.

Now some arrows are also possible with diagonals. For example A (2,5) and B (5,9). If You check length of this arrow, so it is also 5. So we should include such arrows also.

I hope now You are clear.
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Re: Arrow AB which is a line segment exactly 5 units along with [#permalink]
Length of a line segment^2 = (x2-x1)^2 +(y2-y1)^2 =5^2 (exact length is 5 )
Here x1,x2,y1,y2 are integers between 0 and 9(inclusive)

=> (x2-x1) is integer and (y2-y1) is integer .Let x2-x1=a and y2-y1=b

possible cases :

1. a=0,b=5
2.a=5,b=0
3.a=3,b=4
4.a=4,b=3

For above count(s) of possible combinations are :

1.10*5*2 (inverse ,e.g.: (0,9-4) and (0,4-9) ,no doubling where diff is zero to avoid double count )
2.5*2*10
3.7*6*4 (multiplies by 4 since 4 cases possible for each 7*6 combo )
4.6*7*4

Above sums up to 536
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Re: Arrow AB which is a line segment exactly 5 units along with [#permalink]
harikris wrote:
Arrow AB which is a line segment exactly 5 units along with an arrowhead at A is to be constructed in the xy-plane. The x and y coordinates of A and B are to be integers that satisfy the inequalities 0 ≤ x ≤ 9 and 0 ≤ y ≤ 9. How many different arrows with these properties can be constructed ?

A. 50
B. 168
C. 200
D. 368
E. 536

Consider the diagram.
The arrows could be vertical, horizontal or diagonal.

Attachment:
Ques4.jpg

The vertical arrows are shown by the blue arrows. 5 of them will start from x = 0, 5 from x = 1 and so on till x = 9. So you have 50 of these blue arrows. You have another 50 vertical arrows which are the same arrows but with the arrow head on the opposite end (shown by the red arrow). So you have a total of 100 vertical arrows.
Similarly, you have 100 horizontal arrows.

Now check out the diagonal arrows. One co-ordinate should be of length 3 and another of 4 (so that the arrow length is 5 and all points are integers). Look at the purple arrows. The x co-ordinate is 3 and the y co-ordinate is 4. You can make 7*6 = 42 such arrows. Similarly, you can make 42 arrows with x cor-ordinate as 4 and y co-ordinate as 3. So you have 84 arrows. But you get another set of 84 arrows by keeping the arrows the same but putting the arrow head on the opposite end so you get a total of 2*84 = 168 arrows.

Similarly, you can make arrows in the opposite direction shown by the green arrows. So you have another 168 arrows.

Total = 100 + 100 + 168 + 168 = 536

Why we cannot build horizontal arrows?
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Re: Arrow AB which is a line segment exactly 5 units along with [#permalink]
Solving without graph.
Attachments

File comment: Solving without graph.

photo_2019-04-23_18-21-36.jpg [ 115.74 KiB | Viewed 2213 times ]

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Re: Arrow AB which is a line segment exactly 5 units along with [#permalink]
Hello!

We can solve this question by using Combinatorics as well.

My approach for this question is below:

Given two points with their co-ordinates: A(x1, y1), B(x2, y2). Then the length of AB should be:

AB^2 = 5^2 =25 = (x1-x2)^2 + (y1-y2)^2 = X^2 + Y^2 in which I denote X = x1-x2 and Y = y1-y2

Our goal is to determine X and Y such that X^2 + Y^2 = 25. Let's find out all the pairs (X^2, Y^2) that meet such a requirement, and also keep in mind that the square root of Y^2 and X^2 has to be integers. One example below does not meet this requirement such as the pair (1,24) as square root of 24 is not an integer:

(X^2,Y^2) = { (0,25), (1,24), ..., (9,16),...} Feel Free to exploit the whole set but we have found only one candidate in this list that meets the requirements above: (9,16) = (3^2, 4^2)

Next, we need to find out all the pairs (x1, x2) that has a difference of 3 and all the pairs (y1,y2) that has a difference of 4.

1. All the pairs (x1, x2) that has a difference of 3: (0,3), (1,4), ..., (6,9) and you flip the coordinates = 14 pairs
AND all the pairs (y1,y2) that has a difference of 4: (0,4), (1,5),...,(5,9) and you flip the coordinates = 12 pairs

2. Question: How many ways for you to choose a pair from 7 pairs AND a pair from 6 pairs? 14 x 12 = 168
Next: Don't forget, there are 2 ways to arrange 2 chose pairs once we choose one from (y1,y2) and one from (x1,x2) so we have a total of 168 x 2 = 336

3. Do the same thing for special case: AB^2 = 5^2 =25 = (x1-x2)^2 + (y1-y2)^2 = X^2 + Y^2 = 0 + 25
Find All the pairs (x1, x2) that has a difference of 0: (0,0), (1,1), ..., (9,9) and you do not flip the coordinates in this case as you will get identical pairs = 10 pairs
AND all the pairs (y1,y2) that has a difference of 5: (0,5), (1,6),...,(4,9) and you flip the coordinates = 10 pairs

4. Question: How many ways for you to choose a pair from 10 pairs AND a pair from 10 pairs? 10 x 10 = 100
Next: Don't forget, there are 2 ways to arrange 2 chose pairs once we choose one from (y1,y2) and one from (x1,x2) so we have a total of 100 x 2 = 200

The FINAL Answer is: 336 + 200 = 536
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Arrow AB which is a line segment exactly 5 units along with [#permalink]
harikris wrote:
Arrow AB which is a line segment exactly 5 units along with an arrowhead at A is to be constructed in the xy-plane. The x and y coordinates of A and B are to be integers that satisfy the inequalities 0 ≤ x ≤ 9 and 0 ≤ y ≤ 9. How many different arrows with these properties can be constructed ?

A. 50
B. 168
C. 200
D. 368
E. 536

Consider the diagram.
The arrows could be vertical, horizontal or diagonal.

Attachment:
Ques4.jpg

The vertical arrows are shown by the blue arrows. 5 of them will start from x = 0, 5 from x = 1 and so on till x = 9. So you have 50 of these blue arrows. You have another 50 vertical arrows which are the same arrows but with the arrow head on the opposite end (shown by the red arrow). So you have a total of 100 vertical arrows.
Similarly, you have 100 horizontal arrows.

Now check out the diagonal arrows. One co-ordinate should be of length 3 and another of 4 (so that the arrow length is 5 and all points are integers). Look at the purple arrows. The x co-ordinate is 3 and the y co-ordinate is 4. You can make 7*6 = 42 such arrows. Similarly, you can make 42 arrows with x cor-ordinate as 4 and y co-ordinate as 3. So you have 84 arrows. But you get another set of 84 arrows by keeping the arrows the same but putting the arrow head on the opposite end so you get a total of 2*84 = 168 arrows.

Similarly, you can make arrows in the opposite direction shown by the green arrows. So you have another 168 arrows.

Total = 100 + 100 + 168 + 168 = 536

Kindly explain this, please. I was able to narrow down the answer to E since I knew the answer is more than 400 and there was only one option. But, I am curious how you got 168. I am not able to comprehend the logic behind 7 * 6 = 42 onwards.
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Re: Arrow AB which is a line segment exactly 5 units along with [#permalink]
harikris wrote:
Arrow AB which is a line segment exactly 5 units along with an arrowhead at A is to be constructed in the xy-plane. The x and y coordinates of A and B are to be integers that satisfy the inequalities 0 ≤ x ≤ 9 and 0 ≤ y ≤ 9. How many different arrows with these properties can be constructed ?

A. 50
B. 168
C. 200
D. 368
E. 536

There are 2 kinds of arrows: (5,0) and (3,4) arrows
for (5,0) arrows : (9-5+1)*(9-0+1)=50
these are in 4 directions: 4*50=200 arrows
for (3,4) arrows : (9-3+1)*(9-4+1)=42
these are in 8 directions: 8*42=336 arrows
total 200+336=536 arrows
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Re: Arrow AB which is a line segment exactly 5 units along with [#permalink]
KarishmaB wrote:
harikris wrote:
Arrow AB which is a line segment exactly 5 units along with an arrowhead at A is to be constructed in the xy-plane. The x and y coordinates of A and B are to be integers that satisfy the inequalities 0 ≤ x ≤ 9 and 0 ≤ y ≤ 9. How many different arrows with these properties can be constructed ?

A. 50
B. 168
C. 200
D. 368
E. 536

Consider the diagram.
The arrows could be vertical, horizontal or diagonal.

Attachment:
Ques4.jpg

The vertical arrows are shown by the blue arrows. 5 of them will start from x = 0, 5 from x = 1 and so on till x = 9. So you have 50 of these blue arrows. You have another 50 vertical arrows which are the same arrows but with the arrow head on the opposite end (shown by the red arrow). So you have a total of 100 vertical arrows.
Similarly, you have 100 horizontal arrows.

Now check out the diagonal arrows. One co-ordinate should be of length 3 and another of 4 (so that the arrow length is 5 and all points are integers). Look at the purple arrows. The x co-ordinate is 3 and the y co-ordinate is 4. You can make 7*6 = 42 such arrows. Similarly, you can make 42 arrows with x cor-ordinate as 4 and y co-ordinate as 3. So you have 84 arrows. But you get another set of 84 arrows by keeping the arrows the same but putting the arrow head on the opposite end so you get a total of 2*84 = 168 arrows.

Similarly, you can make arrows in the opposite direction shown by the green arrows. So you have another 168 arrows.

Total = 100 + 100 + 168 + 168 = 536

KarishmaB - Should we not consider arrows that are parallel to X axis?
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Re: Arrow AB which is a line segment exactly 5 units along with [#permalink]
Engineer1 wrote:
KarishmaB wrote:
harikris wrote:
Arrow AB which is a line segment exactly 5 units along with an arrowhead at A is to be constructed in the xy-plane. The x and y coordinates of A and B are to be integers that satisfy the inequalities 0 ≤ x ≤ 9 and 0 ≤ y ≤ 9. How many different arrows with these properties can be constructed ?

A. 50
B. 168
C. 200
D. 368
E. 536

Consider the diagram.
The arrows could be vertical, horizontal or diagonal.

Attachment:
Ques4.jpg

The vertical arrows are shown by the blue arrows. 5 of them will start from x = 0, 5 from x = 1 and so on till x = 9. So you have 50 of these blue arrows. You have another 50 vertical arrows which are the same arrows but with the arrow head on the opposite end (shown by the red arrow). So you have a total of 100 vertical arrows.
Similarly, you have 100 horizontal arrows.

Now check out the diagonal arrows. One co-ordinate should be of length 3 and another of 4 (so that the arrow length is 5 and all points are integers). Look at the purple arrows. The x co-ordinate is 3 and the y co-ordinate is 4. You can make 7*6 = 42 such arrows. Similarly, you can make 42 arrows with x cor-ordinate as 4 and y co-ordinate as 3. So you have 84 arrows. But you get another set of 84 arrows by keeping the arrows the same but putting the arrow head on the opposite end so you get a total of 2*84 = 168 arrows.

Similarly, you can make arrows in the opposite direction shown by the green arrows. So you have another 168 arrows.

Total = 100 + 100 + 168 + 168 = 536

KarishmaB - Should we not consider arrows that are parallel to X axis?

We most certainly are to consider them.
We found that we get 100 vertical arrows. So then we will have 100 horizontal arrows too (see the highlighted above). This is so because 0 ≤ x ≤ 9 and 0 ≤ y ≤ 9.
Re: Arrow AB which is a line segment exactly 5 units along with [#permalink]
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