To solve the problem, let's break it down step by step:

Step 1: Understanding the Problem

We have a rectangle with side lengths 10 and 24 units. The vertices of this rectangle are on a circle with a radius of 13 units. We need to find the area between the rectangle and the circle.

Step 2: Calculate the Area of the Rectangle

The area of a rectangle is given by: $\text{Area of the rectangle} = \text{length} \times \text{width}$ Substituting the given values: $\text{Area of the rectangle} = 10 \times 24 = 240 \, \text{square units}$

Step 3: Verify that the Rectangle Fits Inside the Circle

The rectangle is inscribed in the circle. This means the diagonal of the rectangle is equal to the diameter of the circle.

The diagonal $d$ of a rectangle can be found using the Pythagorean theorem: $d = \sqrt{(\text{length})^2 + (\text{width})^2}$ Substituting the given values: $d = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26 \, \text{units}$ The diameter of the circle is twice the radius: $\text{Diameter of the circle} = 2 \times 13 = 26 \, \text{units}$ Since the diagonal of the rectangle is equal to the diameter of the circle, the rectangle can indeed fit inside the circle.

Step 4: Calculate the Area of the Circle

The area of a circle is given by: $\text{Area of the circle} = \pi r^2$ Substituting the given radius: $\text{Area of the circle} = \pi \times 13^2 = \pi \times 169 \approx 531.35 \, \text{square units}$

Step 5: Find the Area Between the Figures

The area between the circle and the rectangle is simply the difference between the area of the circle and the area of the rectangle: $\text{Area between the figures} = \text{Area of the circle} - \text{Area of the rectangle}$ Substituting the values: $\text{Area between the figures} = 531.35 - 240 = 291.35 \, \text{square units}$

Final Answer

The area between the figures is approximately 291.35 square units.

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Would you like more details or have any other questions?

Here are 5 related questions you might find interesting:

1. How would the answer change if the side lengths of the rectangle were different?
2. What if the circle had a different radius but the rectangle's side lengths remained the same?
3. Can we have other polygons with vertices on the same circle? How do we calculate the area between them?
4. How would you find the area of the circle if it wasn't perfectly circular (e.g., an ellipse)?
5. What if the rectangle wasn't inscribed, but rather circumscribed about the circle?

Tip: When dealing with inscribed or circumscribed figures, the Pythagorean theorem is often useful for connecting the geometry of the figure to the properties of the circle.