Coordinate geometry ...

[ketandoshi]

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 ae to be integers that satisfy the inequalities -4<=x<=5 and 6<=y<=16. How many different triangles with these properties could be constructed?



Thanks
Ketan

[CompBanker]

My take on this problem:

I believe the best way to tackle this problem is to determine how many possible places the coordinate "P" can sit on the x-axis and how many possible places the coordinate "Q" can sit on the y-axis. For every possible placement of "P" on the x-axis, we will be able to construct a triangle for every possible placement of "Q" on the y-axis.

Knowing that the x-coordinates of the points on the triangle are bound between -4 and 5, we know that the number of possibilities for P must be:

9 = (5) - (-4) + (1) - (1).

The reason we subtract 1 is because point P and point R can not form a triangle if they sit on the same point, so the last coordinate in the range is reserved for R.

We can use the same methodology for determining all the possible y-coordinates of point "Q."

10 = (16) - (6) + (1) - (1)

So now we know that there are 9 different x-coordinates for point "P" and 10 different y-coordinates for point "Q". Be careful here. Simply multiplying 9 * 10 = 90 will give a wrong answer. You have to take into consideration that point "P" can move up and down the y-axis as well. To compensate for this, we need to multiply the number of x-coordinate possibilities for point "P" by the number of y-coordinate possibilities for point "Q" FOR EVERY y-coordinate possibility for point "P". The formula looks as follows:

9 = Number of x-coordinate possibilities for point "P"

10 = Number of y-coordinate possibilities for point "Q" when point "P" is at y = 6.
9 = Number of y-coordinate possibilities for point "Q" when point "P" is at y = 7.
8 = Number of y-coordinate possibilities for point "Q" when point "P" is at y = 8.
7 = Number of y-coordinate possibilities for point "Q" when point "P" is at y = 9.
etc. etc.

Answer = 9 * (10+9+8+7+6+5+4+3+2+1) = 9*55 = 495

[shibal]

The lowest coordinate point for x can be (-4, 6).

Go horizontally [from -4] to select a point and vertically [from -4] to select another point.

Since you have already chosen -4 as x-coordinate, you have horizontally 9 points [from -3 to + 5 inclusive, i.e., 9 points].
Similarly, since you have chosen 6 as y-coordinate, you have vertically 10 points [from 7 to 16 inclusive, i.e., 10 points].

So, at (-4, 6), you have 10*9 or 90 right triangles.

Now shift the point right to another point, (i.e.-3, 6). Again, you have 9 points; although P now moves by 1 point horizontally, total points do not decrease, because it can now take -4's original position at left also.

Similarly for (-3, 6), it has 10 points vertically. Again, we have 90 triangles.

we can conclude that we have 10 values for x coordinate & 11 values for y-coordinate. Total 10*11=110 points.

No. of total triangles = 110*90 = 9900

[Prashantbhardwaj]

A right angle triangle PQR with right angle P will be formed in a coordinate system only if coordinates of the three points are:

P (x,y) Q(x,y') R(x',y)

i.e. PR and PQ remain straight lines perpendicular to each other

As such the variations possible are:

P for product of total no. of values for both x and y.

Q 1 for x and 1 less values of y for P.

and R 1 for y and 1 less values of x for P.

So the arrangements of coordinates are:

P= 10*11=110 [-4<=x<=5 givning 10 values]
Q= 1*10=10 [6<=y<=16 giving 11 values]
R= 9*1=9

now forming a triangle is an and condition as all points must exist together therefore total no. of triangles is 110*10*90=9900.

[venmic]

I understood all of it except
and R 1 for y and 1 less values of x for P.

how does it work for R

A right angle triangle PQR with right angle P will be formed in a coordinate system only if coordinates of the three points are:

P (x,y) Q(x,y') R(x',y)

i.e. PR and PQ remain straight lines perpendicular to each other

As such the variations possible are:

P for product of total no. of values for both x and y.

Q 1 for x and 1 less values of y for P.

and R 1 for y and 1 less values of x for P.

So the arrangements of coordinates are:

P= 10*11=110 [-4<=x<=5 givning 10 values]
Q= 1*10=10 [6<=y<=16 giving 11 values]
R= 9*1=9

now forming a triangle is an and condition as all points must exist together therefore total no. of triangles is 110*10*90=9900.

[Prashantbhardwaj]

I understood all of it except
and R 1 for y and 1 less values of x for P.

how does it work for R

A right angle triangle PQR with right angle P will be formed in a coordinate system only if coordinates of the three points are:

P (x,y) Q(x,y') R(x',y)

i.e. PR and PQ remain straight lines perpendicular to each other

As such the variations possible are:

P for product of total no. of values for both x and y.

Q 1 for x and 1 less values of y for P.

and R 1 for y and 1 less values of x for P.

So the arrangements of coordinates are:

P= 10*11=110 [-4<=x<=5 givning 10 values]
Q= 1*10=10 [6<=y<=16 giving 11 values]
R= 9*1=9

now forming a triangle is an and condition as all points must exist together therefore total no. of triangles is 110*10*90=9900.

Q 1 for x and 1 less values of y for P.

and R 1 for y and 1 less values of x for P.

Let me clarify it. I skipped a detail.

What I meant to say here is that the value of y for R will be the same as the value of y for P and value of x for Q will also be the same as the value of x for P. As such the variations in x for Q and y for R have been counted for in P.

Thus there is 1 value of x for Q and there is one value of y in R for a given value of P(x,y).

I have attached an image if it helps image.

[Prashantbhardwaj]

And from above since one value of the coordinates of R is already the same as that of P's x so if the other i.e. the value of y is also taken to be same as that of P's y then there will be no triangle as P and R become the same point.

So the variations for R are 10-1 = 9 where the one value less is the value of x for P which R can never have.

And same goes for Q for value of y.
we get 10 variable values for Q