(E) 10
4. In Figure 5 , if the radius of the semicircle is 5 , what is the
area of the inscribed square?

To find the area of the inscribed square in a semicircle with a radius of 5, we can start by determining the side length of the square. When a square is inscribed in a semicircle, the top corners of the square touch the arc of the semicircle.

Let’s denote the side length of the square as “s”. The relation between the side length of the inscribed square and the radius of the semicircle can be derived using the Pythagorean theorem. In this case, the half of the side length of the square and the radius will form a right triangle.

Thus, .

Solving this gives .

You can simplify this to get the value of s, which, when squared gives you the area of the square. Through calculations, you will find that the area of the inscribed square is 25. Therefore, the answer is (D) 25.

Now…
The concept of inscribed shapes is not just a mind-bending math problem but has real-world applications! For instance, architects often use inscribed shapes to optimize space and materials in design. By maximizing an area within given constraints (like a semicircle), you can better understand how to efficiently utilize available resources.

Another fun fact: Geometry and the study of shapes date back to ancient civilizations! The Greeks utilized these ideas in their architecture and art, seeing beauty in proportion and symmetry—after all, that perfect square doesn’t just pop up in math, it’s everywhere in culture too!