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ConfusedFreshy

  • 2 years ago

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!

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  1. ConfusedFreshy
    • 2 years ago
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    |dw:1344217058690:dw|

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

  3. ConfusedFreshy
    • 2 years ago
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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.

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

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

  6. shadowfiend
    • 2 years ago
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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 :/

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

  8. ConfusedFreshy
    • 2 years ago
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    Is that better?

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

  10. ConfusedFreshy
    • 2 years ago
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    I think the central angle is supposed to be 90 degrees. That is why it is there.

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

  12. ConfusedFreshy
    • 2 years ago
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    Yes.

  13. shadowfiend
    • 2 years ago
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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?

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

  15. shadowfiend
    • 2 years ago
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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?

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

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

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

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

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

  21. shadowfiend
    • 2 years ago
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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?

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

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

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

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

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

  27. shadowfiend
    • 2 years ago
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    Correct. So now let's figure out the area of the *unshaded* portion of the sector.

  28. shadowfiend
    • 2 years ago
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    We can do that by finding the area of the triangle formed by the central angle and the chord, right?

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

  30. shadowfiend
    • 2 years ago
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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|

  31. ConfusedFreshy
    • 2 years ago
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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?

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

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

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

  35. ConfusedFreshy
    • 2 years ago
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    You just lost me again. Which part am I looking for? THe rest of the circle?

  36. shadowfiend
    • 2 years ago
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    Yeah, the shaded area of the circle is everything except the unshaded part we just found the area of right?

  37. ConfusedFreshy
    • 2 years ago
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    I seriously do not know. I feel so stupid whenever I do geometry. Is it 8?

  38. shadowfiend
    • 2 years ago
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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?

  39. shadowfiend
    • 2 years ago
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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.

  40. ConfusedFreshy
    • 2 years ago
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    so its 45.699?

  41. shadowfiend
    • 2 years ago
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    Sounds about right!

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

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

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