Question

In circle QQ, the length of \overset{\LARGE\frown}{RS} = \frac{4}{3}\pi RS ⌢ = 3 4 ​ π and m\angle RQS=120^\circ∠RQS=120 ∘ . Find the area shaded below. Express your answer as a fraction times \piπ.

Answer 1

The area of the shaded region can be expressed as 5/6π.

Here's how to find the area of the shaded region in circle Q:

**1. Calculate the area of sector ROS:**

- The central angle of sector ROS is 4/3π radians.

- The area of a sector is calculated as a fraction of the whole circle's area, proportional to the central angle:

Area of sector ROS = (4/3π radians) * πr², where r is the radius of circle Q.

**2. Calculate the area of triangle ROS:**

- Triangle ROS is isosceles because angle ROS is 120° (the sum of the angles in a triangle is 180°, so the other two angles must be 30° each).

- The height of the triangle is drawn from R to the midpoint of QS, which is also the radius of the circle (since ROS is a diameter).

- The base of the triangle is QS, which is also the diameter of the circle.

- Therefore, the area of triangle ROS is:

Area of triangle ROS = (1/2) * base * height = (1/2) * QS * r.

**3. Calculate the area of the shaded region:**

- The shaded region is the difference between the area of sector ROS and the area of triangle ROS:

Area of shaded region = Area of sector ROS - Area of triangle ROS.

**4. Substitute and simplify:**

- Substituting the expressions for the sector and triangle areas:

Area of shaded region = ((4/3π radians) * πr²) - ((1/2) * QS * r)

= (4/3π - 1/2) * πr²

= (8/6 - 3/6) * πr²

= 5/6 * πr²

Therefore, the area of the shaded region can be expressed as 5/6π.

The question probable maybe:

In circle Q, the length of widehat RS= 4/3 π and m∠ RQS=120°. Find the area shaded below. Express your answer as a fraction times π.
Diagram-(Given in the attachment)

Answer 2

We can start by finding the length of the arc \overset{\LARGE\frown}{TS} using the fact that the length of the arc \overset{\LARGE\frown}{RS} is \frac{4}{3}\pi times the radius of the circle. Since angle RQS is 120 degrees, arc \overset{\LARGE\frown}{TS} is \frac{1}{3} of the circumference of the circle:

arc \overset{\LARGE\frown}{TS} = \frac{1}{3} (2\pi R) = \frac{2}{3}\pi R

Next, we can find the length of the chord TS using the Law of Cosines:

TS^2 = TR^2 + RS^2 - 2(TR)(RS)\cos(\angle TRS)

Since \angle TRS is 120 degrees, we have:

TS^2 = R^2 + (4/3)^2R^2 - 2(R)(4/3)R(-1/2)

Simplifying this expression, we get:

TS^2 = \frac{25}{9}R^2

Taking the square root of both sides, we get:

TS = \frac{5}{3}R

Now we can find the height of the shaded region by drawing the altitude from the center of the circle to chord TS. This altitude bisects chord TS and is also perpendicular to it, so it divides TS into two segments of equal length:

Height = \frac{1}{2}(TS) = \frac{5}{6}R

Finally, we can find the area of the shaded region by subtracting the area of triangle RST from the area of sector RQS:

Area of sector RQS = (120/360)\pi R^2 = \frac{1}{3}\pi R^2

Area of triangle RST = (1/2)(RS)(height) = (1/2)(4/3)R(\frac{5}{6}R) = \frac{5}{9}R^2

Area of shaded region = Area of sector RQS - Area of triangle RST = \frac{1}{3}\pi R^2 - \frac{5}{9}R^2 = \frac{2}{9}\pi R^2

Therefore, the area shaded below is \frac{2}{9}\pi times the square of the radius R of the circle.