Tough question-1

[\'manpreet singh]

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) 1,100
(C) 9,900
(D) 10,000
(E) 12,100

[GMATGuruNY]

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

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 P, Q and R to form a triangle. For each point, we need to choose an x value and a y value.

Point P:
x value: -4≤x≤5, giving us 10 choices.

y value: 6≤y≤16, giving us 11 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 10 shirts and 11 ties, and we need to determine how many outfits can be made:

(number of choices for x)*(number of choices for y)=10*11=110 choices for P.

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

y value: If P and Q 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 11 choices for y when we chose P, so we have 11-1=10 choices for Q's y value.

(number of choices for x)*(number of choices for y)=1*10=10 choices for Q.

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

x value: If P and R 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 10 choices for x when we chose P, so we have 10-1=9 choices for R's x value.

(number of choices for x)*(number of choices for y)=9*1=9 choices for R.

So we have 110 choices for P, 10 choices for Q, and 9 choices for R. We need to determine how many ways we can combine P, Q and R to make a triangle. It's as though we have 110 shirts, 10 ties, and 9 pairs of pants, and we need to determine the number of outfits that can be made:

(number of choices for P)*(number of choices for Q)*(number of choices for R) = 110*10*9 = 9900.

The correct answer is C.

[\'manpreet singh]

Thanks Mitch:)

Singh

[sid128]

Here we are talking about a right triangle whose two sides are parallel to x and y axis.

One side is parallel to y axis =>> Two vertices will have same x coordinate.
One side is parallel to X axis =>> Two vertices will have same y coordinate.

Number of x coordinates = 10*9 = 90
Number of y coordinates = 11*10 = 110

Total number of coordinates possible = 110*90 = 9900 i.e C

PS: I took almost 4 minutes to solve this problem

[anjalimanas]

We need to determine the number of possible ways to construct line PR and PQ.

(1)Number of ways to construct line PR is to select pairs of x-coordinate from -4 <=x<=5 = 10 * 9 = 90

(2)Number of ways to construct line PQ is to select pairs of y-coordinate from 6 <=x<=16 = 11 * 10 = 110

Total ways to construct traingle = (1) * (2) = 90 * 110 = 9900



(Since PR is || to x-axis and PQ is || to y axis, We do not have to consider any combinations where the traingle is tilted at any angle from the axis)