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 $y=1/2x+3/2$.