Here is the questions that I am stuck on,

1. Point Y lies on line segment \(\overline{XZ}\) such that XY = 2 and YZ = 4. Semicircles are constructed with diameters \(\overline{XY}, \overline{XZ}\), and \(\overline{YZ}\). Find the area of the blue region.

I dont even know how or where to start.

2. A sector of a circle has a central angle of 100\(^\circ\). If the area of the sector is 250 \(\pi\), what is the radius of the circle?

I try to put my thoughts and ideas into words because I want to learn, but what I tried has no matamatical words.

If someone could help me with these two questions that would be great,

Thanks in advance

Guest Jun 7, 2020

#3**+2 **

1. The distance XZ = XY + YZ = 6, so the radius of the largest semicircle is 3. Hence its area is (1/2)pi3^{2} or A1 = 9pi/2.

The area of the next largest semicircle is A2 = (1/2)pi2^{2} or A2 = 2pi

The area of the smallest semicircle is A3 = (1/2)pi1^{2} or A3 = pi/2

Hence the area of the shaded region = A1 - A2 - A3 = 9pi/2 - 2pi - pi/2 = 2pi

Alan Jun 7, 2020

#4**0 **

1)

The area for a circle is πr^2.

Thus the large circle is π((2+4)/2)^2 /2 = 9π/2.

The smaller ones are 1π/2 = π/2 and 4π/2 = 2π. Thus we have 9π/2 - π/2 - 2π = 4π-2π = 2π. as the area of blue.

2)

The formula for circle are is πr^2. You have 100/360 = 50/180 = 5/18 part of the area.

5/18 * πr^2 = 250π

5πr^2/ 18 = 250 π

5r^2/18 = 250

5r^2 = 250 * 18

r^2 =250*18/5 = 50*18 = 900

r = 30

If you don't understand anything feel free to ask!

hugomimihu Jun 7, 2020