Problem Solving

Please help me with a simpler solution

Refer Attachment

A rectangle is inscribed in a circle of radius r. If the rectangle is not a square, which of the following could be equal to the perimeter of the rectangle?

A. 2r(Sqrt 3)
B. 2r(Sqrt 3 + 1)
C. 4r(Sqrt 2)
D. 4r(Sqrt 3)
E. 4r(Sqrt3 + 1)

When shapes overlap, look for what they have IN COMMON.
When a rectangle is inscribed in a circle, the DIAGONAL of the rectangle is also the DIAMETER of the circle.

Almost every answer choice here includes √3.
√3 implies a 30-60-90 triangle.
The sides of a 30-60-90 triangle are in the following ratio:
1 - √3 - 2.

Let r=1, implying a diameter of 2.
Draw the following figure:



The perimeter of the rectangle above = 2 + 2√3. This is our target.
Now we plug r=1 into the answers to see whether one of them yields our target of 2 + 2√3.

Answer choice B:
2r(√3 + 1) = 2(1)(√3 + 1) = 2√3 + 2.
Success!

The correct answer is B.