Question

If point \( A(x,2) \) is equidistant from the points \( B(8,-2) \) and \( C(2,-2) \), find the value of \( x \).

Hint:

Distance Formula: \( AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Answer 1

Using the distance formula:
\[ AB^2 = AC^2 \] \[ (x-8)^2 + (2+2)^2 = (x-2)^2 + (2+2)^2 \] \[ (x-8)^2 + 16 = (x-2)^2 + 16 \] Cancel 16 from both sides:
\[ (x-8)^2 = (x-2)^2 \] \[ x^2 - 16x + 64 = x^2 - 4x + 4 \] \[ -16x + 64 = -4x + 4 \] \[ -12x = -60 \] \[ x = 5 \] Thus, \( x = \mathbf{5} \).
Correct Answer: \( x = 5 \)