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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # Right triangle PQR is to be constructed in the xy-plane so that the ri

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Intern  Joined: 22 Dec 2009
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Right triangle PQR is to be constructed in the xy-plane so that the ri  [#permalink]

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95 00:00

Difficulty:   55% (hard)

Question Stats: 64% (02:06) correct 36% (02:37) wrong based on 1157 sessions

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Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis. The x and Y coordinates of P,Q and R are to be integers that satisfy the inequalities -4≤ X≤ 5 and 6≤ y≤ 16. How many different triangles with these properties could be constructed?

A. 110
B. 1100
C. 9900
D. 10000
E. 12100
Math Expert V
Joined: 02 Sep 2009
Posts: 60647
Re: Right triangle PQR is to be constructed in the xy-plane so that the ri  [#permalink]

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12
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gmatJP wrote:
Can anyone please tell me how to figure this out...

Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis. The x and Y coordinates of P,Q and R are to be integers that satisfy the inequalitites -4≤ X≤ 5 and 6≤ y≤ 16. How many different triangles with these properties could be constructed?

A)110
B)1100
C)9900
D)10000
E)12100

thanks..

We have the rectangle with dimensions 10*11 (10 horizontal dots and 11 vertical). PQ is parallel to y-axis and PR is parallel to x-axis.

Choose the (x,y) coordinates for vertex P (right angle): 10C1*11C1;
Choose the x coordinate for vertex R (as y coordinate is fixed by A): 9C1, (10-1=9 as 1 horizontal dot is already occupied by A);
Choose the y coordinate for vertex Q (as x coordinate is fixed by A): 10C1, (11-1=10 as 1 vertical dot is already occupied by A).

10C1*11C1*9C1*10C1=9900.

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Re: Right triangle PQR is to be constructed in the xy-plane so that the ri  [#permalink]

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10
8
imo C
total values for x=10;y=11

x1,y1=10*11.......................coordinates of 1st pnt
x2,y2=9*1(y2=y1)............... coordinates of 2nd pnt y coordinates will be same as that of 1st pnt bcoz it is parallel to x axis
x3,y3=1*10(x2=x3)..........coordinates of 3rd pt. x coordinates will be same as that of 2nd point bcoz to make a right angle it has to be parallel to y axis

tot ways=10*11*9*1*1*10=9900

oa pls
##### General Discussion
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Re: Right triangle PQR is to be constructed in the xy-plane so that the ri  [#permalink]

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1
Hello,
Pls explain.

x1,y1=10*11.......................coordinates of 1st pnt
x2,y2=9*1(y2=y1)............... coordinates of 2nd pnt y coordinates will be same as that of 1st pnt bcoz it is parallel to x axis
x3,y3=1*10(x2=x3)..........coordinates of 3rd pt. x coordinates will be same as that of 2nd point bcoz to make a right angle it has to be parallel to y axis

you mentioned, y2=y1. but calculation value is different. same for x2 and x3.
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Re: Right triangle PQR is to be constructed in the xy-plane so that the ri  [#permalink]

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First a picture to summarize the discussion so far.
Attachment: Triangles.JPG [ 8.26 KiB | Viewed 21351 times ]

I disagree with 10 possible x values for P, because the length of PR must be an integer

P(X1,Y1)

X1 can be {-4,4} but not 5
Y1 can be {6,15} but not 15

I actually come up with 8100 triangles the regular way

And then a question

What in the question excludes a triangle like this, and why is this triangle not considered different?

Attachment: Whynot.JPG [ 3.51 KiB | Viewed 21341 times ]
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Re: Right triangle PQR is to be constructed in the xy-plane so that the ri  [#permalink]

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Quote:
Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis. The x and Y coordinates of P,Q and R are to be integers that satisfy the inequalitites -4≤ X≤ 5 and 6≤ y≤ 16. How many different triangles with these properties could be constructed?

A)110
B)1100
C)9900
D)10000
E)12100

First: I see these four basic triangle orientations. I do not see why the question does not consider them.
Attachment: 4triangles.JPG [ 14.25 KiB | Viewed 21407 times ]

Each triangle shows PR parallel to the x-axis
Each triangle shows the right angle at P

Second: If we exclude all the triangles except the one in the upper right. I keep coming up with only 8100 possible triangles

Because Point P and Point R share a y value, they must both be integers, and they must be different (the triangles must have 3 sides)
Because Point P and Point Q share an x value, they must both be integers, and they must be different (the triangles must have 3 sides)
We know Point P coordinates are ( -4≤ X≤ 4 , 6≤ y≤ 15)

Because we know that each x and each y must be integers, the two legs of the triangle have limited possible lengths

1 ≤ length PR ≤ 9

1 ≤ length PQ ≤ 10

With this in mind I find only 8100 possible triangles resulting from ( 90 coordinate pairs possible for P times 9 lengths for PR times 10 lengths for PQ = 90 * 9 * 10)
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7
3
2 more hours and I have it figured out.

P (A,B)
R (C,B)
Q (A,D)

So you need four values to solve the problem.

2 x-values {A and C}

2 y-values {B and D}

When you approach the question this way the orientation of the triangle is irrelevant. Because by using the same x-value for point P and point Q along with the same y-value for point point P and point R. The right angle is at P and PR is parallel to the x-axis.

So I need 2 x values from -4≤ X≤ 5.
Pick the first one from 10 choices
Pick the second one from 9 choices

10*9 = 90

and I need 2 y values from 6≤ y≤ 16
Pick the first one from 11 choices
Pick the second one from 10 choices

11*10 = 110

So combine the 2 and we have 9900 (110*90) possible triangles.

I realized my error when I started to make a table of the possible coordinates for point P using the triangle in the upper right of the four choices

With point P and (4,6) I realized that I only have one possible length for PR
*If P is (4,6)
and
the triangle has the orientation in the upper right of my image.
*Then R must be (5,6)

That got me back to the starting line and my wonderful Chinese girlfriend helped me going in the correct direction.

She is junior in college and an English major by the way, with no math since high school. Retired Moderator Joined: 20 Dec 2010
Posts: 1533
Re: Right triangle PQR is to be constructed in the xy-plane so that the ri  [#permalink]

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2
Official answer is "C" - 9900

This was a hard question for me.

Anyway, I took the following approach to solve this.

****************************************

When y=6
We know -4<=x<=+5, there are 10 integral points.
The points are: {-4,6},{-3,6},{-2,6},{-1,6},{0,6},{1,6},{2,6},{3,6},{4,6},{5,6}
So, number of distinct line segments that can be formed are 10C2=45.
And these line segments PR as it should be parallel to x-axis.
With every line segment there can be 20 triangles:

Let's see how:

consider line segment with points: P={-4,6} and R={1,6}
We know there are 10 points above P that can serve us as Q. Q should always be
vertically above or below point P because P is the right angle.
So Q can be: {-4,7},{-4,8},{-4,9},{-4,10},{-4,11},{-4,12},{-4,13},{-4,14},{-4,15},{-4,16}
See for yourself that now: for just one segment PQ we have 10 triangles.
PQR.
However, we can flip the points on the same line segment PR, so that P={1,6} and R={-4,6}
We know there are 10 points above P that can serve us as Q. Q should always be
vertically above or below point P because P is the right angle.
So in this case Q can be:
{1,7},{1,8},{1,9},{1,10},{1,11},{1,12},{1,13},{1,14},{1,15},{1,16}
See for yourself that for just one segment PQ we have another 10 triangles.
So, a total of 20 distinct triangles for just one line segment.

thus: a total of 20 * 10C2 = 20 * 45 = 900 triangles for all line segments where PR
is on y=6.

Remember, we were talking only about PR that lies on y=6 line.

There are 11 such lines between y=6 and y=16 i.e. 16-6+1.

Thus; a total of 900 * 11 = 9900 triangles can be formed with given conditions.
*******************************************************************************

~fluke
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Re: Right triangle PQR is to be constructed in the xy-plane so that the ri  [#permalink]

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1
1
In the figure shown

When P is fixed , q can take 10 positions( remaining 10 co-ordinates up the y axis) and R can take 9 positions ( x-axis) .
Hence 10*9 = 90 traingles
and p itself can take - 10 *11 positions =110
hence 90*110 = 9900
Attachments traingle.png [ 4.86 KiB | Viewed 17534 times ]

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Re: Right triangle PQR is to be constructed in the xy-plane so that the ri  [#permalink]

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1
there are 10 points on the x axis from which we can choose any twos and then form the desired PR base.

By the same way ,there are 11 points on the y-axis from which we can choose any twos and then form the desired PQ perpendicular.

We can form 10P2 * 11P2=9900
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This was a very hard question for me.

I know that the set of vertices that should satisfy the inequalities apply to all P,Q and R. I took a look at the options and noticed only C that could be divisible by 3.. since set of all the vertices/3. Hence picked C.
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Re: Right triangle PQR is to be constructed in the xy-plane so that the ri  [#permalink]

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Please Bunuel , remove the 'work-rate problems' tag
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Posts: 60647
Re: Right triangle PQR is to be constructed in the xy-plane so that the ri  [#permalink]

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Mahmoudfawzy83 wrote:
Please Bunuel , remove the 'work-rate problems' tag

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Done. Thank you.
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Re: Right triangle PQR is to be constructed in the xy-plane so that the ri  [#permalink]

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lets say the coordinates of P, Q, R are (x1,y1), (x1,y2), (x2,y1).

Now we want (x1,x2) combinations * (y1,y2) combinations
x1,x2 & y1,y2 can never be equal to form a triangle
and they both satisfy
-4≤ X≤ 5 and 6≤ y≤ 16.

x1,x2 combinations = 10 * 10 (total combinations) - 10(both are same) = 90
y1,y2 combinations = 11 * 11 (total combinations) - 11(when both are same) = 110

Hence total number of combinations = 90 * 110 = 9900

A perfect example of permutations and combinations with Co-ordinate geometry.

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Re: Right triangle PQR is to be constructed in the xy-plane so that the ri  [#permalink]

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gmatJP wrote:
Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis. The x and Y coordinates of P,Q and R are to be integers that satisfy the inequalities -4≤ X≤ 5 and 6≤ y≤ 16. How many different triangles with these properties could be constructed?

A. 110
B. 1100
C. 9900
D. 10000
E. 12100

total x coordinates= 10
y coordinates= 11
so
for P we can have = 10c1*11c1
since its a right angled triangle so point Q would be fixed and value of x =1 and y can be choosen in 10c1 ways
for point R , value of x would be 1 less than 10 ; 9c1 and y would be fixed so 1
10c1*11c1*10c1*9c1
= 9900
IMO C Re: Right triangle PQR is to be constructed in the xy-plane so that the ri   [#permalink] 02 Feb 2019, 00:39
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