To find the length of the diameter of the circle in which a rectangle is inscribed, given that the area of the rectangle is 1,089√3 square units, and the rectangle's diagonal is twice the length of its shortest side, we will proceed step-by-step.
1. Identify Variables and Relationships:
- Let the length of the shortest side of the rectangle be [tex]\(a\)[/tex].
- Let the length of the other side be [tex]\(b\)[/tex].
- Given that the diagonal is twice the length of the shortest side, the diagonal is [tex]\(2a\)[/tex].
2. Area of the Rectangle:
- The area of the rectangle is given as [tex]\(1089\sqrt{3}\)[/tex] square units.
- Therefore, we have the relationship: [tex]\(a \cdot b = 1089\sqrt{3}\)[/tex].
3. Using the Pythagorean Theorem:
- The rectangle's diagonal forms a right triangle with its sides [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
- According to the Pythagorean theorem, the relationship between the sides and the diagonal of a rectangle is: [tex]\(a^2 + b^2 = (2a)^2\)[/tex].
- Simplifying this gives: [tex]\(a^2 + b^2 = 4a^2\)[/tex].
- Further simplification yields: [tex]\(b^2 = 3a^2\)[/tex].
4. Substitute and Simplify:
- We know [tex]\(b = \sqrt{3}a\)[/tex] from the equation [tex]\(b^2 = 3a^2\)[/tex].
- Substitute [tex]\(b\)[/tex] back into the area equation:
[tex]\[
a \cdot (\sqrt{3}a) = 1089\sqrt{3}
\][/tex]
- This simplifies to:
[tex]\[
a^2\sqrt{3} = 1089\sqrt{3}
\][/tex]
- By dividing both sides by [tex]\(\sqrt{3}\)[/tex], we get:
[tex]\[
a^2 = 1089
\][/tex]
- Solving for [tex]\(a\)[/tex]:
[tex]\[
a = \sqrt{1089} = 33
\][/tex]
5. Determine the Length of the Other Side ([tex]\(b\)[/tex]):
- From the relationship [tex]\(b = \sqrt{3}a\)[/tex]:
[tex]\[
b = \sqrt{3} \cdot 33 \approx 57.16
\][/tex]
6. Calculate the Diameter of the Circle:
- The diameter of the circle is the same as the diagonal of the rectangle.
- Since the diagonal is [tex]\(2a\)[/tex]:
[tex]\[
\text{Diameter} = 2 \cdot 33 = 66
\][/tex]
Thus, the length of the diameter of the circle is 66 units.
