1. The problem is to understand how to draw internal and external tangents to two circles. 2. First, let's define the terms: - Internal tangents are lines that touch both circles and cross the line segment joining the centers. - External tangents are lines that touch both circles but do not cross the line segment joining the centers. 3. To draw these tangents, you need the centers and radii of the two circles, say $C_1$, $C_2$ with radii $r_1$, $r_2$. 4. The distance between centers is $d = |C_1C_2|$. 5. For external tangents, the length of the tangent segment between the points of tangency is given by $$\sqrt{d^2 - (r_1 - r_2)^2}$$ 6. For internal tangents, the length is $$\sqrt{d^2 - (r_1 + r_2)^2}$$ 7. To construct the tangents: - Draw the line segment $C_1C_2$. - For external tangents, construct circles with radii $r_1$ and $r_2$ and find points where lines tangent to both circles do not cross $C_1C_2$. - For internal tangents, find lines tangent to both circles that cross $C_1C_2$. 8. The exact construction involves geometric methods or algebraic equations of circles and lines, but the key is understanding the difference in how the tangent lines relate to the segment $C_1C_2$. This explanation helps you visualize and draw internal and external tangents correctly.