Find the Area of the Inscribed Rectangle

Hi Eftball, let's solve this geometry puzzle. We need to find the area of the blue rectangle inscribed in this semicircle.

Let's analyze the given diagram. We have a semicircle with a diameter. The problem shows that the distance from the left edge to a point on the diameter is ten.

Wait, the diagram shows the 'ten' starts from the left edge and ends at the vertex of the rectangle. Let's define the radius of the semicircle as r.

Looking closely at the rectangle, we see tick marks indicating that the side lengths are in a specific ratio. The long side is twice the length of the short side.

Let's place the semicircle on a coordinate plane with the center at the origin zero zero. The equation of the semicircle is x squared plus y squared equals r squared for y greater than or equal to zero.

Let the rectangle's vertex on the base be the point P. Let the rectangle be tilted. However, there is a simpler way. Notice symmetry and the midpoints.

Looking at the properties, let the width of the rectangle be a. Then the length is two a.

Note that one vertex of the rectangle is on the diameter, and two other vertices touch the circular arc. Let's call the vertex on the diameter the origin for a temporary coordinate shift.

Because the rectangle is inscribed like this, the distance labeled ten is actually the distance from the edge of the semicircle to the point where the rectangle touches the diameter.

Let's denote the coordinates of the vertices. If we rotate the figure so the rectangle is axis-aligned, it becomes clearer. In the original orientation, the length of the diagonal of the rectangle is related to the diameter.