Right triangle PQR is to be constructed in the xy- plane so that the right angle is at P and line 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

Question

Right triangle PQR is to be constructed in the xy- plane so that the right angle is at P and line 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 could be constructed with these properties?a)110b)1100c)9900d)10000Correct answer is option 'C'. Can you explain this answer?

Answer

In order to solve this problem, we need to consider the properties of a right triangle and the given inequalities.

A right triangle has one angle equal to 90 degrees, so the right angle at point P satisfies this condition.

Line PR is parallel to the x-axis, which means that the y-coordinate of point R must be the same as the y-coordinate of point P. Let's call this y-coordinate 'y'.

Now, let's consider the inequalities. We are given that -4 ≤ x ≤ 3 and -3 ≤ y ≤ 5. We need to find integer coordinates that satisfy these inequalities.

Since line PR is parallel to the x-axis, the x-coordinate of point R must be the same as the x-coordinate of point Q. Let's call this x-coordinate 'x'.

Now, let's consider the right triangle PQR. We know that the length of line PR is equal to the difference between the x-coordinates of P and R, which is |x - x| = 0. Since line PR is parallel to the x-axis, it is a horizontal line with length 0.

The length of line PQ is equal to the difference between the y-coordinates of P and Q, which is |y - y| = 0. Since line PQ is parallel to the y-axis, it is a vertical line with length 0.

Therefore, the right triangle PQR degenerates into a single point at P, with coordinates (x, y).

In order to satisfy the given inequalities, we need to find integer values of x and y that fall within the given ranges. Since -4 ≤ x ≤ 3 and -3 ≤ y ≤ 5, we can choose any integer values for x and y that fall within these ranges. For example, we can choose x = 0 and y = 0, which satisfies the inequalities and gives us a right triangle with coordinates P(0, 0), Q(0, 0), and R(0, 0).

Therefore, the integer coordinates of P, Q, and R that satisfy the given conditions are P(0, 0), Q(0, 0), and R(0, 0).

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