Here are the step-by-step constructions for the two geometry problems shown in the image: --- Problem 1: Construct an Interior Tangent to Two Circles of the Same Diameter Given: Diameter of both circles = $30 \text{ mm}$ (so, radius $R = 15 \text{ mm}$) Distance between their centers = $120 \text{ mm}$ Construction Steps: 1. Draw the Center Line: Draw a horizontal line. Mark two points, O and O', on this line such that the distance between them is $120 \text{ mm}$. These will be the centers of the two circles. 2. Draw the Circles: With O as the center, draw a circle with a radius of $15 \text{ mm}$. With O' as the center, draw another circle with a radius of $15 \text{ mm}$. 3. Find the Midpoint: Find the midpoint M of the line segment OO'. This can be done by drawing arcs of equal radius (greater than half of OO') from O and O', and connecting their intersection points. The line connecting these points will bisect OO' at M. 4. Draw the Semicircle: With M as the center, draw a semicircle with radius MO (or MO'). This semicircle will pass through O and O'. 5. Locate Tangent Points on Construction Circle: From O, draw a line perpendicular to OO' upwards. This line will intersect the semicircle at a point, let's call it X. From O', draw a line perpendicular to OO' downwards. This line will intersect the semicircle at a point, let's call it Y. 6. Draw the Interior Tangent: Draw a straight line connecting point X and point Y. This line XY is the common interior tangent to the two circles. --- Problem 2: Construct an External Tangent Between Two Circles of Different Radii Given: Radius of the first circle $R_1 = 25 \text{ mm}$ Radius of the second circle $R_2 = 15 \text{ mm}$ Distance between their centers = $100 \text{ mm}$ Construction Steps: 1. Draw the Center Line: Draw a horizontal line. Mark two points, O and O', on this line such that the distance between them is $100 \text{ mm}$. O will be the center of the larger circle, and O' for the smaller one. 2. Draw the Main Circles: With O as the center, draw a circle with radius $R_1 = 25 \text{ mm}$. With O' as the center, draw a circle with radius $R_2 = 15 \text{ mm}$. 3. Draw the Difference Circle: Calculate the difference in radii: $R_{diff} = R_1 - R_2 = 25 \text{ mm} - 15 \text{ mm} = 10 \text{ mm}$. With O as the center, draw a third (construction) circle with radius $R_{diff} = 10 \text{ mm}$. 4. Find the Midpoint: Find the midpoint M of the line segment OO'. 5. Draw the Semicircle: With M as the center, draw a semicircle with radius MO (or MO'). This semicircle will pass through O and O'. 6. Locate Tangent Point on Difference Circle: From O', draw a line perpendicular to OO' that intersects the semicircle at a point, let's call it P. 7. Draw the Tangent to the Difference Circle: Draw a straight line from O to P. This line OP is tangent to the difference circle (the $10 \text{ mm}$ radius circle). 8. Locate Tangent Point on Larger Circle: Extend the line OP until it intersects the larger circle (radius $R_1 = 25 \text{ mm}$) at a point. Let's call this point T1. This is the tangent point on the larger circle. 9. Locate Tangent Point on Smaller Circle: From O', draw a line parallel to the line segment OT1. This parallel line will intersect the smaller circle (radius $R_2 = 15 \text{ mm}$) at a point. Let's call this point T2. This is the tangent point on the smaller circle. 10. Draw the External Tangent: Draw a straight line connecting point T1 and point T2. This line T1T2 is the common external tangent to the two circles.