haidgmat 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
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.
Hope this helps!
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at
[email protected].
Student Review #1
Student Review #2
Student Review #3
