knight247 wrote:Right triangle ABC is to be drawn in the xy-plane so that the right angle is at A and AB is parallel to the y-axis. If the x- and y-coordinates of A, B, and C are to be integers that are consistent with the inequalities -6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?
A. 54
B. 432
C. 2160
D. 2916
E. 148,824
OA C. Detailed explanations would be appreciated.
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 conditions given.
For each point, we need to choose an x value and a y value.
Point A:
x value: -6≤x≤2, giving us 9 choices.
y value: 4≤y≤9, giving us 6 choices.
To combine our choices for x with our choices for y, we multiply:
9*6 = 54.
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 6 choices for y when we chose A, so we have 6-1 = 5 choices for C's y value.
To combine our choices for x with our choices for y, we multiply:
1*5 = 5.
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.
To combine our choices for x with our choices for y, we multiply:
8*1 = 8.
Now we need to count how many ways A, B and C can be combined to form a triangle.
To combine our choices for A with our choices for B and C, we multiply:
54*5*8 = 2160.
The correct answer is
C.
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