If the distance between \( (x, 5) \) and \( (1,2) \) is 5 , where \( x>0 \), then \( x \) is equal to

To find the value of \( x \), we start by using the distance formula between the points \( (x, 5) \) and \( (1, 2) \). The distance \( d \) is given by: \[ d = \sqrt{(x - 1)^2 + (5 - 2)^2} \] We know the distance is 5, so we set up the equation: \[ \sqrt{(x - 1)^2 + (3)^2} = 5 \] Squaring both sides, we have: \[ (x - 1)^2 + 9 = 25 \] Subtracting 9 from both sides gives: \[ (x - 1)^2 = 16 \] Taking the square root of both sides, we find: \[ x - 1 = 4 \quad \text{or} \quad x - 1 = -4 \] Thus, \[ x = 5 \quad \text{or} \quad x = -3 \] Since \( x > 0 \), we discard \( x = -3 \). Finally, we find: \[ \boxed{5} \]