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
+33615 1. The distance XZ = XY + YZ = 6, so the radius of the largest semicircle is 3. Hence its area is (1/2)pi32 or A1 = 9pi/2.
The area of the next largest semicircle is A2 = (1/2)pi22 or A2 = 2pi
The area of the smallest semicircle is A3 = (1/2)pi12 or A3 = pi/2
Hence the area of the shaded region = A1 - A2 - A3 = 9pi/2 - 2pi - pi/2 = 2pi
+1005 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!
