Question

A semicircle has a center at X and diameter WZ. The radius XY of the semicircle has a length of 2. The chord YZ has a length of 2. What is the area shaded sector formed by angle WXY

Answer 1

To find the area of the shaded sector formed by angle WXY, first find the angle measure of WXY using the properties of a central angle. Then use the formula for the area of a sector, A = (θ/360) × π × r^2, to calculate the area.

Step-by-step explanation:

To find the area of the shaded sector formed by angle WXY, we first need to find the angle measure of WXY. Since XY is the radius and YZ is the chord of the semicircle, angle WXY is the central angle of the semicircle. The measure of a central angle is equal to twice the measure of the inscribed angle.

The length of XY is 2, and the length of YZ is 2. Since the radius and the chord are congruent, this means that angle XYZ is also 2. Therefore, angle WXY is twice that, which is 4.

Now, to find the area of the shaded sector, we use the formula for the area of a sector: A = (θ/360) × π × r^2. Plugging in the values, we get A = (θ/360) × π × 2r^2 = (1/90) × 4π = (2π/45) square units.

Answer 2

, it is important to recognize that the point (x,y) in the first quadrant represents any point on the semicircle. In Figure 2, the same semicircle is shown with the inscribed rectangle drawn for three different values of x.