AB is the hypotenuse of the triangle which has vertices of (0/0), (3/0) and (3/4). The lengths of the two shorter sides are 3 (horizontal side) and 4 (vertical side). This is the most common form of a 3 - 4 - 5 Pythagorean triple. Thus the hypotenuse is has a length of 5.

Twice the line segment equals a distance of 10. So every point which has a distance equal to 10 to point B is a possible coordinate. Since the point can be anywhere with a distance of 10, we can think of the set of possible places for point C as all points on the "outer line of" a circle with a radius of 10.

Circle formula: (x-h)^2 + (y-k)^2 = r^2 ,where (h/k) is the center of the circle and r is the radius. In this case, the center is (3/4) and the radius is 10.

Thus we plug in the given values in the formula.

(x-3)^2 + (y-4)^2 = 10^2

Finally, we plug in the x and y coordinates of the answer choices in the formula.

A) (-5-3)^2 + (-2-4)^2 = 10^2
64 + 36 =100

...

C) (10-3)^2 + (11-4)^2 = 10^2
49 + 49 =/= 100
Here, the equation is not fulfilled and thus the distance from B to C is shorter than 10.

Any other solutions for this question?

THanks, this solution is by far the best way to see it I believe