Help with triangle question

[islandgurl918]

Right triangle LMN is to be constructed in the xy-plane so that the right angle is at point L and LM is parallel to the x-axis. The x- and y- coordinates of L, M, and N are to be integers that satisfy the inequalities -3 < x < 4 and 3 < y < 11. How many different triangles with these properties could be constructed?


(A) 72

(B) 576

(C) 4032

(D) 4608

(E) 6336

[GMATGuruNY]

Right triangle LMN is to be constructed in the xy-plane so that the right angle is at point L and LM is parallel to the x-axis. The x- and y- coordinates of L, M, and N are to be integers that satisfy the inequalities -3 < x < 4 and 3 < y < 11. How many different triangles with these properties could be constructed?


(A) 72

(B) 576

(C) 4032

(D) 4608

(E) 6336

When a question asks for the number of triangles that can be constructed, it's not a geometry question but a combinations question. Why? Because a triangle is a combination of 3 points.

We need to determine how many ways we can combine L, M and N to form a triangle. For each point, we need to choose an x value and a y value.

Point L:
x value: -3≤x≤4, giving us 8 choices.

y value: 3≤y≤11, giving us 9 choices.

Now we have to combine the number of choices for x with the number of choices for y. It's as though we have 8 shirts and 9 ties, and we need to determine how many outfits can be made:

(number of choices for x)*(number of choices for y)= 8*9 = 72 choices for L.

Point N:
x value: In order to construct a right triangle, N must have the same x coordinate as L (so that N is directly above L and we get a right angle). So we have only 1 choice for x: it must be the same integer that we chose for N's x value.

y value: If L and N have the same x value, they can't have the same y value, or they will be the same point. We used 1 of our 9 choices for y when we chose L, so we have 9-1=8 choices for N's y value.

(number of choices for x)*(number of choices for y)=1*8=8 choices for N.

Point M:
y value: For LM to be parallel to the x axis, L and M must share the same y value. So the number of choices for y is 1; it must be the same integer that we chose for L's y value.

x value: If L and M have the same y value, they can't have the same x value, or they will be the same point. We used 1 of our 8 choices for x when we chose L, so we have 8-1=7 choices for M's x value.

(number of choices for x)*(number of choices for y)=1*7=7 choices for M.

So we have 72 choices for L, 8 choices for N, and 7 choices for M. We need to determine how many ways we can combine L, N and M to make a triangle. It's as though we have 72 shirts, 8 ties, and 7 pairs of pants, and we need to determine the number of outfits that can be made:

(number of choices for L)*(number of choices for N)*(number of choices for M) = 72*8*7 = 4032.

The correct answer is C.

Hope this helps!

[islandgurl918]

Thank you this is very helpful, sadly better than the Knewton explanation.

[Rahul@gurome]

This is same as asking that in how many ways a right angled triangle can be constructed such that one side is parallel to x axis and the x and y coordinates satisfy the inequality -3<=x<=4 and 3<=y<=11. Also these values of x and y have to be integers.

Let Sx = {-3, -2, -1, 0, 1, 2, 3, 4}.
Let Sy = {3, 4, 5, 6, 7, 8, 9, 10, 11}.

If we select 2 points from Sx say x1, x2 and 2 points from Sy say y1 and y2, we can get two lines: (1) joining (x1,y1) and (x2,y2)
(2) joining (x1, y2) and (x2, y1).
Take line (1) . You can make two right angled triangles on it with one side parallel to x axis.
(a) M(x1, y1), L(x2, y1), N(x2, y2).
(b) N(x1, y1), L(x1, y2), M(x2, y2).
Take line (2). You can make two right angled triangles on it with one side parallel to x axis.
(a) M(x1, y2), L(x2, y2), N(x2, y1)
(b) N(x1, y2), L(x1, y1), M(x2, y1).

So if we select 2 points from Sx and 2 points from Sy, we can construct 4 triangles with it satisfying the above given conditions.

Number of ways of selecting 2 points from Sx and 2 from Sy is 8C2 * 9C2 .

So the number of possible triangles with given conditions that can be constructed is 8C2 * 9C2 * 4 = 4032.

[goyalsau]

x value: -3≤x≤4, giving us 7 spaces. Between two points on the number line
x value: -3≤x≤4, giving us 8 choices. In total points on the number line.


y value: 3≤y≤11, giving us 8 spaces. Between two points on the number line
y value: 3≤y≤11, giving us 9 choices. In total points on the number line.

Then 7 * 8 * 8 * 9 = 4032

You can apply this with any question on this sort answer will be correct always......... ( especially in right angels. )
[/b]

[vijchid]

The original question had the following constraint

-3 < x < 4 and 3 < y < 11.

How come all of you decided to change the constraint -3<=x<=4 and 3<=y<=11?

[goyalsau]

The original question had the following constraint

-3 < x < 4 and 3 < y < 11.

How come all of you decided to change the constraint -3<=x<=4 and 3<=y<=11?

You are right , I didn't notice at all I have seen one same question in OG and there its equal that's why may be,
but ya you are right it was a big mistake,
but now the problem is even more

answer is 5 * 6 * 6 * 7 = 1260

and we don't have that option.......... :roll: