To solve this, let's break down the problem:

• We have a quarter circle with center at point B.
• The rectangle ABCD fits within the quarter circle and touches the circle at point D.
• Points B and C are at the center and on the edge of the quarter circle, respectively.
• The diagonal we need to find is AC, and we know BC is 5 cm, as indicated by the diagram.

Steps to Solve:

1. Identify the radius of the circle: Since B is at the center of the circle and BC is 5 cm (the length from the center to the edge of the circle), the radius of the circle is 5 cm.

2. Use the Pythagorean theorem in triangle ABC: Triangle ABC is a right triangle, with BC as one leg, AB as the other leg (which is the radius of the circle, so 5 cm), and AC as the hypotenuse (the diagonal we want to find).

   The Pythagorean theorem states that: $AC^2 = AB^2 + BC^2$ Substituting the known values: $AC^2 = 5^2 + 5^2 = 25 + 25 = 50$ Taking the square root of both sides: $AC = \sqrt{50} = 5\sqrt{2} \approx 7.07 \text{ cm}$

So, the length of the diagonal AC is approximately 7.07 cm.

Would you like further clarification or more details on any step?

Here are some related questions that may interest you:

1. How do you find the area of a quarter circle?
2. What other geometric properties can be found using the Pythagorean theorem?
3. How can the relationship between a rectangle and a circle be used in geometry problems?
4. What happens if the rectangle is not touching the circle at point D but somewhere else?
5. How would this problem change if the circle were a full circle instead of a quarter circle?

Tip: When working with right triangles, always consider applying the Pythagorean theorem to solve for unknown sides.