Find the equations of the lines that pass through the point (1,2) and are tangent to the circle with equation \(x^2 + y^2 = 25\).

Step 1 ：The equation of the circle is \(x^2 + y^2 = 25\), the center of the circle is at the origin (0,0) and the radius is 5.

Step 2 ：The equation of the line which passes through the point (1,2) and the center of the circle (0,0) can be found using the slope formula, which is \((y_2 - y_1)/(x_2 - x_1)\). Substituting the points, we get the slope of the line as \(2 - 0)/(1 - 0) = 2\).

Step 3 ：The equation of the line is then \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. Substituting the slope and the point (1,2), we get \(y - 2 = 2(x - 1)\) or \(y = 2x\).

Step 4 ：The lines that are tangent to the circle and pass through the point (1,2) are perpendicular to the line \(y=2x\). The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. Hence the slopes of the tangent lines are \(-1/2\).

Step 5 ：The equations of the tangent lines are then \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. Substituting the slopes \(-1/2\) and the point (1,2), we get the equations of the tangent lines as \(y - 2 = -1/2(x - 1)\) and \(y - 2 = 1/2(x - 1)\), simplifying these we get \(y = -1/2x + 5/2\) and \(y = 1/2x + 3/2\).