Two concentric circles are centered at point O. The inner circle has a radius of 6 units, … — GRE General Test

Two concentric circles are centered at point $O$ . The inner circle has a radius of $6$ units, and the outer circle has a radius of $6\sqrt{3}$ units. Radii $OA$ and $OB$ of the outer circle form a central angle $\angle AOB = 60^\circ$ and intersect the inner circle at points $C$ and $D$ , respectively. Which of the following statements must be true regarding the region and boundary lengths defined by these figures? Select all that apply.

The area of the region bounded by arc $AB$ , arc $CD$ , segment $AC$ , and segment $BD$ is equal to $12\pi$ square units.Answer
The straight-line distance between points $A$ and $B$ is $6\sqrt{3}$ units.Answer
The area of the circular segment bounded by chord $AB$ and minor arc $AB$ is $18\pi - 27\sqrt{3}$ square units.Answer
D
The ratio of the length of arc $CD$ to the length of arc $AB$ is $1 : 3$ .
E
The perimeter of the region bounded by arc $AB$ , arc $CD$ , segment $AC$ , and segment $BD$ is $2\pi(1 + \sqrt{3}) + 6\sqrt{3}$ units.

Answer

The true statements are: the area of the region bounded by arc AB, arc CD, segment AC, and segment BD is 12π square units; the straight-line distance between points A and B is 6√3 units; and the area of the circular segment bounded by chord AB and minor arc AB is 18π - 27√3 square units.

The area of the region between the two concentric arcs is obtained by subtracting the inner sector area (6π) from the outer sector area (18π), giving 12π square units. The triangle OAB is equilateral because it has two sides of length 6√3 and an included angle of 60°, making chord AB equal to 6√3. Subtracting the area of this equilateral triangle (27√3) from the outer sector area (18π) yields the area of the circular segment bounded by chord AB and arc AB, which is 18π - 27√3.

Step-by-Step Solution

Calculate the area of sector OAB and sector OCD to evaluate the area of the annular sector region.

Sector OAB area = (60/360) * π * (6√3)^2 = (1/6) * 108π = 18π. Sector OCD area = (60/360) * π * (6^2) = (1/6) * 36π = 6π. Region area = 18π - 6π = 12π.

The area between two concentric sector arcs bounded by the same radii is the difference in sector areas.

Determine the length of chord AB using triangle properties.

In triangle OAB, OA = OB = 6√3 and angle AOB = 60°. An isosceles triangle with a 60° vertex angle is equilateral, so AB = 6√3.

All internal angles of an isosceles triangle with a 60° angle must equal 60°.

Calculate the area of circular segment AB.

Area of equilateral triangle OAB = (√3 / 4) * (6√3)^2 = 27√3. Segment area = Sector OAB area - Triangle OAB area = 18π - 27√3.

A circular segment's area is found by subtracting the area of the subtended triangle from the area of the corresponding sector.

Evaluate arc length ratio and total perimeter of the annular sector.

Arc CD / Arc AB = 6 / (6√3) = 1 / √3 ≠ 1 / 3. Total perimeter = Arc AB + Arc CD + 2*(R - r) = 2√3π + 2π + 2*(6√3 - 6) = 2π(1 + √3) + 12√3 - 12.

Verifies that statements regarding ratio 1:3 and incomplete perimeter calculations are mathematically false.

Key Concept

Concentric circle geometry, arc length proportions, sector area calculations, and circular segment area formulation.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic