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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

 Jun 7, 2020
 #1
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1. The area of the shaded region is 4*pi.

 

2. The radius of the circle is 25.

 Jun 7, 2020
 #2
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hmmm I am still confused... I dont understand how you got those answers nor they make sence... I am sorry but these answers with no explantions did not help me...

Guest Jun 7, 2020
 #3
avatar+30325 
+2

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

 Jun 7, 2020
 #4
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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!

 Jun 7, 2020
edited by hugomimihu  Jun 7, 2020
 #5
avatar+30325 
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2. Area of circle A = pi.r2    Area of circle/Area of segment = A/250pi.  But this must equal 360/100.

 

I'll leave you to take it from here.

 

I see hugo... has already done it!
 

 Jun 7, 2020
edited by Alan  Jun 7, 2020
 #6
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WOW!! thx! I read through your guys explantions and I understand now! I was really stuck on those... 

Thank you agian!!

 Jun 7, 2020

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