Problem Solving

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.2,160
D.2,916
E.148,824

Hi, i guess (A) is the answer for this problem,
we can select the x's value in 9 ways and y's value in 6 ways, since we are given the angle
between these two sides, the third side can be drawn only in one way, which depends on the values
we take for x and y.
so 9x6=54

what is OA

rijul007 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.2,160
D.2,916
E.148,824

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: -6≤x≤2, giving us 9 choices.

y value: 4≤y≤9, giving us 6 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 6 ties, and we need to determine how many outfits can be made:

(number of choices for x)*(number of choices for y) = 9*6 = 54 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 6 choices for y when we chose A, so we have 6-1 = 5 choices for B's y value.

(number of choices for x)*(number of choices for y) = 1*5 = 5 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 54 choices for A, 5 choices for C, and 8 choices for B. We need to determine how many ways we can combine A, C and B to make a triangle. It's as though we have 54 shirts, 5 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 C)*(number of choices for B) = 54*5*8 = 2160.

The correct answer is C.

OA is C