Question

A rhombus is inscribed in a rectangle which in turn is inscribed in a circle as shown in the figure below. P is the centre of all three shapes, PQ=QR=5 units. What is the perimeter of the rhombus?

Answer 1

Correct Answer: 40

To determine the perimeter of the rhombus inscribed in the rectangle, which is inscribed in a circle, consider the given information:

Step 1: Recognize the setup  The rhombus is inside a rectangle, which in turn is inscribed in a circle. Point P is the center, and PQ=QR=5 units, meaning P is the center of both rhombus and rectangle.

Step 2: Understand the circle's diameter Since the rectangle is inscribed in the circle, its diagonal is the circle's diameter.

Step 3: Calculate the rectangle's dimensions using the chord property PQ and QR being equal at 5 units means triangle PQR is an isosceles right triangle (45°-45°-90° triangle) due to inscribed angle properties. Use the Pythagorean theorem to find the rectangle's half-diagonal: \(d = \sqrt{PQ^2 + QR^2} = \sqrt{5^2 + 5^2} = \sqrt{50} = 5\sqrt{2}\) Thus, the circle's radius R is 5√2, and its diameter is \(10\sqrt{2}\).

Step 4: Analyze the implications for the rhombus The rhombus's diagonals coincide with the rectangle's edges. Both diagonals are equal to diameters of the circle. A rhombus with diagonals that are equal implies it is also a square.

Step 5: Calculate the rhombus's (square's) side length Considering the diameter, the calculation is: \(a = \frac{d}{\sqrt{2}} = \frac{10\sqrt{2}}{\sqrt{2}} = 10\)

Step 6: Determine the perimeter of the rhombus The perimeter of the square is calculated as: \(4 \times 10 = 40\)

Validation The computed perimeter of 40 lies within the required range of (40, 40).