cuty wrote: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
got it from knewton practice test
Given the answer choices, I suspect that the ranges of the x and y coordinates are supposed to include the endpoints: -3 ≤ x ≤ 4 and 3 ≤ y ≤ 11.
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 L'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) = 7*1 = 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, M and N 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.
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