The problem presented involves a rectangle inscribed in a circle. Here's a breakdown of the given information:

• A rectangle is inscribed in a circle, meaning each vertex of the rectangle touches the circumference of the circle.
• The diagonal of the rectangle is twice the length of the shortest side of the rectangle.
• The area of the rectangle is given as $1,089 \sqrt{3}$ square units.
• The goal is to find the length of the diameter of the circle.

Let's break this down step by step:

1. Rectangle Inscribed in a Circle: If a rectangle is inscribed in a circle, the diagonal of the rectangle is the diameter of the circle. Let's denote the length of the shorter side of the rectangle as $a$ and the longer side as $b$. The diagonal $d$ can be expressed using the Pythagorean theorem: $d = \sqrt{a^2 + b^2}$ Since $d$ is the diameter of the circle, our goal is to find $d$.

2. Given: Diagonal is twice the length of the shorter side: We are told that the diagonal is twice the length of the shortest side, so: $d = 2a$

3. Given: Area of the Rectangle: The area of the rectangle is $A = ab$, and we are given that: $A = 1,089 \sqrt{3}$ So: $ab = 1,089 \sqrt{3}$

4. Use of Diagonal: From the diagonal equation $d = 2a$, we can also write: $d^2 = a^2 + b^2$ Substituting $d = 2a$ into this: $(2a)^2 = a^2 + b^2$ $4a^2 = a^2 + b^2$ $3a^2 = b^2$ $b = \sqrt{3}a$

5. Substitute into the Area Equation: Now substitute $b = \sqrt{3}a$ into the area equation $ab = 1,089 \sqrt{3}$: $a \cdot \sqrt{3}a = 1,089 \sqrt{3}$ $\sqrt{3}a^2 = 1,089 \sqrt{3}$ Cancel $\sqrt{3}$ from both sides: $a^2 = 1,089$ $a = \sqrt{1,089} = 33$

6. Find the Diameter: Since $d = 2a$, the diameter of the circle is: $d = 2 \times 33 = 66 \, \text{units}$

Final Answer:

The diameter of the circle is 66 units.

Do you have any questions, or would you like further details on any part of the solution?

Here are 5 related questions to expand your understanding:

1. What is the relationship between the diagonal of a rectangle and the circle in which it is inscribed?
2. How does the Pythagorean theorem apply to the sides of a rectangle?
3. What are some examples of real-life objects that can be modeled as inscribed shapes in circles?
4. How would the answer change if the diagonal was a different multiple of the shorter side?
5. Can a square be inscribed in a circle, and if so, what would its properties be?

Tip: Always ensure that relationships like those between the diagonal and sides of a shape are checked carefully to avoid errors in geometric problems.