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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!

  • 2 years ago
  • 2 years ago

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

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

    • 2 years 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.

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

    • 2 years ago
  5. ConfusedFreshy Group Title
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    I believe it is the central angle.

    • 2 years 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 :/

    • 2 years ago
  7. ConfusedFreshy Group Title
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    |dw:1344217343489:dw|

    • 2 years ago
  8. ConfusedFreshy Group Title
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    Is that better?

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

    • 2 years 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.

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

    • 2 years ago
  12. ConfusedFreshy Group Title
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    Yes.

    • 2 years 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?

    • 2 years ago
  14. ConfusedFreshy Group Title
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    Yes I know that.

    • 2 years 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?

    • 2 years ago
  16. ConfusedFreshy Group Title
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    4, since the radius is 4.

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

    • 2 years ago
  18. ConfusedFreshy Group Title
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    What I got could not be correct.

    • 2 years ago
  19. shadowfiend Group Title
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    What did you get?

    • 2 years 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

    • 2 years 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?

    • 2 years ago
  22. ConfusedFreshy Group Title
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    Yes. Which would equal 180?

    • 2 years 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?

    • 2 years ago
  24. ConfusedFreshy Group Title
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    Yes. That divided by 4 is 4.

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

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

    • 2 years 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.

    • 2 years 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?

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

    • 2 years 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|

    • 2 years 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?

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

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

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

    • 2 years 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?

    • 2 years 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?

    • 2 years 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?

    • 2 years 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?

    • 2 years 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.

    • 2 years ago
  40. ConfusedFreshy Group Title
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    so its 45.699?

    • 2 years ago
  41. shadowfiend Group Title
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    Sounds about right!

    • 2 years ago
  42. ConfusedFreshy Group Title
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    Thank you so much!!

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

    • 2 years ago
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