Let's break down the problem systematically.

Given:

• A rectangle is inscribed in a circle of radius 8.
• One vertex of the rectangle is located at point $P = (x, y)$ in quadrant 1 on the circle.

Analysis:

1. Circle Equation: The equation of a circle with radius 8 and centered at the origin is: $x^2 + y^2 = r^2$ where $r = 8$, so: $x^2 + y^2 = 64.$

2. Rectangle Symmetry: Since the rectangle is inscribed in the circle, the other three vertices of the rectangle will be symmetric to $P = (x, y)$:

   • $(-x, y)$ in quadrant 2,
   • $(-x, -y)$ in quadrant 3,
   • $(x, -y)$ in quadrant 4.

3. Rectangle Dimensions: The length and width of the rectangle can be determined based on the coordinates of the vertices:

   • Length (horizontal distance): $2x$,
   • Width (vertical distance): $2y$.

4. Area of the Rectangle: The area of the rectangle is the product of its length and width: $\text{Area} = 2x \cdot 2y = 4xy.$

Summary:

• The point $P = (x, y)$ satisfies $x^2 + y^2 = 64$, since it lies on the circle of radius 8.
• The area of the rectangle is $4xy$.

Would you like further details, such as finding the maximum area of the rectangle or discussing the geometric properties in more depth?

5 Related Questions:

1. What is the maximum area that can be enclosed by a rectangle inscribed in a circle?
2. How do you derive the dimensions of a rectangle inscribed in a circle with a given radius?
3. What is the relationship between the angle subtended by the rectangle and the circle’s radius?
4. How can you apply trigonometry to solve problems involving inscribed shapes?
5. What is the general formula for the area of an inscribed rectangle in a circle?

Tip: Maximizing the area of a rectangle inscribed in a circle often involves using trigonometric identities for a more efficient solution.