Let's go through each of these geometry problems one by one, analyzing what is required and finding the values for the unknowns (like $x$, $y$, or $z$) in each figure.

Problem 35:

• Given a triangle that circumscribes a circle, we see angles and segments labeled $x$, $y$, and $z$.
• The angle given outside is $35^\circ$.
• Since we have a circle inscribed in a triangle, and with a given angle, we may need to use properties of tangents and specific circle-triangle relationships. Further clarification may be required to solve this.

Problem 36:

• The problem involves a circle with a chord and a tangent line.
• There is an angle inscribed between $x$ and $y$, with a small angle marked on the tangent.
• If the angle is given, you may need to use the tangent-secant theorem or properties involving angles with a tangent and chord in a circle to determine $x$ and $y$.

Problem 37:

• This problem shows a square around a circle.
• Dimensions of the square and circle are provided, such as $G$, $K = 8$ cm, and $O = 16$ cm. The circle has a radius $6$ cm, and the diameter $D$ to $R = 12$ cm.
• You can use the dimensions to relate the square's side length to the circle's diameter or radius.

Problem 38:

• This is a square with an inscribed circle.
• The radius of the circle is given as $14$ ft.
• You would need to find the side length of the square, which is twice the radius.

Problem 39:

• Another triangle circumscribing a circle, with an angle of $40^\circ$.
• This setup resembles Problem 35, with points marked $x$, $y$, and $z$.
• By using circle-tangent angle properties, you can determine relationships among $x$, $y$, and $z$.

Problem 40:

• A circle with two tangents extending to a triangle with a 40° angle.
• You might need to apply tangent-segment properties and angle relationships within the circle.

Problem 41:

• A right triangle tangent to a circle, with lengths $4$, $8$, and unknown $x$.
• Use the Pythagorean theorem or other tangent relationships.

Problem 42:

• Another right triangle and a circle, with lengths $5$, $12$, and unknown $x$.
• This problem is similar to Problem 41 and may be solved with the Pythagorean theorem.

Would you like to go through each problem in detail, or is there a specific question among these you'd like help solving?

Here are some related questions:

1. How do tangent properties help in solving circle problems with triangles?
2. What is the tangent-secant theorem, and how does it apply here?
3. How can inscribed angles help in determining unknown lengths?
4. What are the properties of a square circumscribing a circle?
5. How can the Pythagorean theorem assist in right triangle-circle problems?

Tip: Always look for symmetry and known angle relationships in circle and triangle geometry to simplify complex problems.