euro wrote:
[2] A right triangle ABC has to be constructed in the xy-plane so that the right angle is at A and AB is parallel to x-axis. The coordinates of A, B and C are to satisfy the inequalities -3 ≤ x ≤ 5 and 2 ≤ y ≤ 11 and x & y are integers. The number of different triangles that can be constructed with these properties are?
(A) 90
(B) 900
(C) 6480
(D) 8100
(E) 10000
Official answer to [2] [spoiler](C) 6480 [/spoiler]
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 A, B and C to form a triangle that satisfies all the requirements of the problem. For each point, we need to choose an x value and a y value.
Point A:
x value: -3≤x≤5, giving us 9 choices.
y value: 2≤y≤11, giving us 10 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 9 shirts and 10 ties, and we need to determine how many outfits can be made:
(number of choices for x)*(number of choices for y) = 9*10 = 90 choices for A.
Point C:
x value: In order to construct a right triangle, C has to have the same x coordinate as A (so that C is directly above A and we get a right angle). So we have only 1 choice for x: it must be the same integer that we chose for A's x value.
y value: If A and C share the same x value, they can't have the same y value, or they will be the same point. We used 1 of our 10 choices for y when we chose A, so we have 10-1 = 9 choices for B's y value.
(number of choices for x)*(number of choices for y) = 1*9 = 9 choices for C.
Point B:
y value: For AB to be parallel to the x axis, A and B 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 A's y value.
x value: If A and B share the same y value, they can't have the same x value, or they will be the same point. We used 1 of our 9 choices for x when we chose A, so we have 9-1 = 8 choices for B's x value.
(number of choices for x)*(number of choices for y)= 8*1 = 8 choices for B.
So we have 90 choices for A, 9 choices for B, and 8 choices for B. We need to determine how many ways we can combine A, B and C to make a triangle. It's as though we have 90 shirts, 9 ties, and 8 pairs of pants, and we need to determine the number of outfits that can be made:
(number of choices for A)*(number of choices for B)*(number of choices for C) = 90*9*8 = 6480.
The correct answer is
C.
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
