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Math Help - Area

  1. #1
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    Area

    Hi all.

    I've been working on a rather simply worded problem that has turned out to be quite a nightmare.

    Problem:

    A semicircle of radius 10 cm has diameter AB. AP and AQ are chords of lengths 6 cm and 8 cm respectively. Calculate the area bounded by these two chords and the arc PQ.

    I've drawn the picture to go with this but that's pretty much it. Can't take it further.

    Any and all help will be greatly appreciated.

    Thanks.
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  2. #2
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    Quote Originally Posted by pollardrho06 View Post
    Hi all.

    I've been working on a rather simply worded problem that has turned out to be quite a nightmare.

    Problem:

    A semicircle of radius 10 cm has diameter AB. AP and AQ are chords of lengths 6 cm and 8 cm respectively. Calculate the area bounded by these two chords and the arc PQ.

    I've drawn the picture to go with this but that's pretty much it. Can't take it further.

    Any and all help will be greatly appreciated.

    Thanks.
    Hi pollardho06,

    You have 2 alternative geometric representations.

    Cord AP could be above the diameter AB along with AQ,
    or AP could be above AB and AQ could be below.

    Suppose they are both above the diameter AB.

    Draw the radius from the centre O to Q.
    You may calculate any angle in the isosceles triangle AQO
    using the Law of Cosines (Cosine Rule).

    This allows you to calculate angle QOB.
    Hence you can calculate the area of sector QOB.

    You can also calculate the area of triangle AQO.

    Now draw the radius from P to the circle centre O.

    The area between the arc AP and the triangle APO can be found similarly.

    Subtract the 3 areas from the area of the semicircle for the final answer.

    That's one way.
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  3. #3
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    Smile

    Quote Originally Posted by Archie Meade View Post
    Hi pollardho06,

    You have 2 alternative geometric representations.

    Cord AP could be above the diameter AB along with AQ,
    or AP could be above AB and AQ could be below.

    Suppose they are both above the diameter AB.

    Draw the radius from the centre O to Q.
    You may calculate any angle in the isosceles triangle AQO
    using the Law of Cosines (Cosine Rule).

    This allows you to calculate angle QOB.
    Hence you can calculate the area of sector QOB.

    You can also calculate the area of triangle AQO.

    Now draw the radius from P to the circle centre O.

    The area between the arc AP and the triangle APO can be found similarly.

    Subtract the 3 areas from the area of the semicircle for the final answer.

    That's one way.
    Great!! Thank you so much Archie. I get the area approx. 2.66 sq. cm
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  4. #4
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    area

    Quote Originally Posted by pollardrho06 View Post
    Great!! Thank you so much Archie. I get the area approx. 2.66 sq. cm
    sq. cm

    Hi pollardrho

    A few comments. After looking at this problem before looking at replies I solved it and got an answerof 2.63 sq.cm. I did not consider an alternate solution because the problem refers only to a semicircle.i solved for angles
    by using the sin of sector angles1/2 chord over radius.


    bjh
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  5. #5
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    Quote Originally Posted by bjhopper View Post
    sq. cm

    Hi pollardrho

    A few comments. After looking at this problem before looking at replies I solved it and got an answerof 2.63 sq.cm. I did not consider an alternate solution because the problem refers only to a semicircle.i solved for angles
    by using the sin of sector angles1/2 chord over radius.


    bjh
    Hi bjh.

    Yes, I also think that since they mention a semicircle there is only one geometric representation of the problem. Your answer is also correct. I used complete values for all intermediate results to get 2.66 in the end.

    Regards,

    Shahz.
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