Shouldn't we consider about how the line AC is parallel to x-axis then it's slope has to be 0 as well. Can we ignore this and still reach at the correct solution?

Coordinates of point A and C will have same y coordinates only, so the slope will be 0 only

For example, when A is (-4, 7) then C can be (-3, 7) or (-2, 7) or (-1, 7) or (0, 7), ..... or (6, 7)
All will give you zero slope.

IMO, right triangles can be drawn with 10*8 number of ways for its (x,y) coordinates as the two different opposing angles may form the right angles.

would it be possible to provide a figure with the explanation please

hi, after reading all replies, I still not quite get it. Could anyone provide explanation on how to solve this question?

Thank you

factor58, pattypat
It would be very difficult to provide eloborated figure but I hope this will help you to grab the concept!

To find out the number of terms (in case of inclusive), we can use the formulae = Last term - First term + 1

Since, \(-4 ≤ x ≤ 6\)
\(x = 6 - (-4) + 1\)
Number of possible values of \(x = 11\)

Also, \(7 ≤ y ≤ 15\)
\(y = 15 - 7 + 1\)
Number of possible values of \(y = 9\)

To costruct a right triangle - 2 points must have same x-coordinate (A and C) and 2 points must have same y-coordinate (A and B).

For example, in all Black triangles:
The Vertical line AB has same x-coordinate as -4 and horizontal lines AC, AC', AC'', .... AC will have same y-coordinate as 7.

Number of ways of picking the x-coordinate of point A out of 11 possible options = \(^{11}C_1 = 11\)
Number of ways of picking the y-coordinate of point A out of 9 possible options = \(^{9}C_1 = 9\)

Since, we have already picked the x and y coordinates of point A - we will now have \(11 - 1 = 10\) possible options for the x-coordinate of point C and \(9 - 1 = 8\) possible options for the y-coordinate of point B

Finally,
Number of ways of picking the x-coordinate of point C = \(^{10}C_1 = 10\)
NOTE: The y-coordinate of point C has to be same as that of Point A to form a Base (Horizontal line) i.e. \(7\)

And,
Number of ways of picking the y-coordinate of point B = \(^{8}C_1 = 8\)
NOTE: The x-coordinate of point B has to be same as that of Point A to form a Perpendicular (Vertictal line) i.e. \(-4\)

Total number of ways to form a right triangle = \((11)(9)(10)(8) = 7920\)

Col. A: \(7920\)
Col. B: \(8000\)

Col. A < Col. B
Hence, option B

KarunMendiratta
Thank you so much

Much obliged, good sir.

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