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ConfusedFreshy Group Title

Find the area of the shaded region. Point O marks the center of the circle. Picture below! I need someone to give me the equation to follow to solve the problem please!

  • one year ago
  • one year ago

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  1. ConfusedFreshy Group Title
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    |dw:1344217058690:dw|

    • one year ago
  2. shadowfiend Group Title
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    Is the entire circle shaded?

    • one year ago
  3. ConfusedFreshy Group Title
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    No the right hand corner arc is not. That is kind of hard to see but it is not shaded up there.

    • one year ago
  4. shadowfiend Group Title
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    Got it. And what's the 90 degrees?

    • one year ago
  5. ConfusedFreshy Group Title
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    I believe it is the central angle.

    • one year ago
  6. shadowfiend Group Title
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    Hm. I feel like I still don't have a good grasp of what the image looks like. Is there any chance you could either take a picture of it and upload it or draw it in a tool that can do the shading a little better and upload that image? We know our drawing tool needs a bit of work :/

    • one year ago
  7. ConfusedFreshy Group Title
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    |dw:1344217343489:dw|

    • one year ago
  8. ConfusedFreshy Group Title
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    Is that better?

    • one year ago
  9. shadowfiend Group Title
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    Definitely, though the 90 degrees still seems somewhat unrelated to anything in the drawing…

    • one year ago
  10. ConfusedFreshy Group Title
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    I think the central angle is supposed to be 90 degrees. That is why it is there.

    • one year ago
  11. shadowfiend Group Title
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    Meaning the angle formed by the two endpoints of the chord and the center, yes?

    • one year ago
  12. ConfusedFreshy Group Title
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    Yes.

    • one year ago
  13. shadowfiend Group Title
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    Ok, then I think what you can do is, because it's a 90 degree angle you know that the circle portion in question is 1/4 the circle, right?

    • one year ago
  14. ConfusedFreshy Group Title
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    Yes I know that.

    • one year ago
  15. shadowfiend Group Title
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    And in order for that to be true, what does the length of the line segment from the center of the circle to the endpoint of the chord have to be?

    • one year ago
  16. ConfusedFreshy Group Title
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    4, since the radius is 4.

    • one year ago
  17. shadowfiend Group Title
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    Precisely. Can you get the area of that sector of the circle, ignoring the chord?

    • one year ago
  18. ConfusedFreshy Group Title
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    What I got could not be correct.

    • one year ago
  19. shadowfiend Group Title
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    What did you get?

    • one year ago
  20. ConfusedFreshy Group Title
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    I got 720. Seeing that the formula for area of a sector is Area= [angle]radius^2/ 2

    • one year ago
  21. shadowfiend Group Title
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    I'll leave aside whether that formula is correct or not, in this particular case we've already determined that this is a quarter of the circle, so really it's just area / 4, right?

    • one year ago
  22. ConfusedFreshy Group Title
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    Yes. Which would equal 180?

    • one year ago
  23. shadowfiend Group Title
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    Well, area is pi * r^2, which here is pi * 4^2, which is pi * 16 right?

    • one year ago
  24. ConfusedFreshy Group Title
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    Yes. That divided by 4 is 4.

    • one year ago
  25. shadowfiend Group Title
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    Right, so we have pi * 4, which is ~12.566

    • one year ago
  26. ConfusedFreshy Group Title
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    What does that mean? Is that the area of the whole sector?

    • one year ago
  27. shadowfiend Group Title
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    Correct. So now let's figure out the area of the *unshaded* portion of the sector.

    • one year ago
  28. shadowfiend Group Title
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    We can do that by finding the area of the triangle formed by the central angle and the chord, right?

    • one year ago
  29. ConfusedFreshy Group Title
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    Ummm. Pythagorean theorem so it could be 32= c^2

    • one year ago
  30. shadowfiend Group Title
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    Well, that's how you'd find the length of the chord. We're just interested in the area of this: |dw:1344218326720:dw|

    • one year ago
  31. ConfusedFreshy Group Title
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    Omg. I just derped so hard there. The base and height of the triangle is 4 and 4. so it would be 8?

    • one year ago
  32. shadowfiend Group Title
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    Exactly! So, how do we get the area of the unshaded area of that sector?

    • one year ago
  33. ConfusedFreshy Group Title
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    12.566 minus 8. 4.566 is the unshaded area?

    • one year ago
  34. shadowfiend Group Title
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    Yep. And now the area of the shaded part of the circle?

    • one year ago
  35. ConfusedFreshy Group Title
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    You just lost me again. Which part am I looking for? THe rest of the circle?

    • one year ago
  36. shadowfiend Group Title
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    Yeah, the shaded area of the circle is everything except the unshaded part we just found the area of right?

    • one year ago
  37. ConfusedFreshy Group Title
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    I seriously do not know. I feel so stupid whenever I do geometry. Is it 8?

    • one year ago
  38. shadowfiend Group Title
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    Well, it's going to be pi*r^2 of the circle - the area we just calculated. Do you see how that works?

    • one year ago
  39. shadowfiend Group Title
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    Basically, to find the area of the circle except for that portion, we worked backwards to find the area of that portion, then subtract it from the area of the circle as a whole.

    • one year ago
  40. ConfusedFreshy Group Title
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    so its 45.699?

    • one year ago
  41. shadowfiend Group Title
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    Sounds about right!

    • one year ago
  42. ConfusedFreshy Group Title
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    Thank you so much!!

    • one year ago
  43. shadowfiend Group Title
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    No problem! Hope that helped the reasoning for other problems!

    • one year ago
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