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Math Help - Circle Geometry

  1. #1
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    Circle Geometry 2

    Points A,B and C lie on a circle. The length of the chord AB is a constant k. Let angle ACB = a degrees and angle ABC = b degrees.

    i) Why is a degrees always constant?

    ii) How does the sum of the lengths of the chords AC and BC become (k/sin a)(sin b + sin(a + b))?

    iii) When b = 90 - a/2, what is the expression for S?

    Could someone please help me (especially with ii)?

    Thanx very much
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  2. #2
    Member Nacho's Avatar
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    i) because <a always takes the same arc AB

    ii) using "sin theorem"

    <br />
\frac{{\sin a}}<br />
{K} = \frac{{\sin b}}<br />
{{AC}} \Leftrightarrow AC = \frac{{K\sin b}}<br />
{{\sin a}}<br />

    again

    <br />
\frac{{\sin a}}<br />
{K} = \frac{{\sin \left( {180 - (a + b)} \right)}}<br />
{{BC}} \Leftrightarrow \frac{{\sin a}}<br />
{K} = \frac{{\sin \left( {a + b} \right)}}<br />
{{BC}} \Leftrightarrow BC = \frac{{K\sin \left( {a + b} \right)}}<br />
{{\sin a}}<br />

    suming

    <br />
AC + BC = \frac{{K\sin b}}<br />
{{\sin a}} + \frac{{K\sin \left( {a + b} \right)}}<br />
{{\sin a}} = \frac{K}<br />
{{\sin a}}\left( {\sin b + \sin \left( {a + b} \right)} \right)<br />

    What is S?
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  3. #3
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    Oops sori

    S = sum of the lengths of the chords AC and BC
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  4. #4
    Member Nacho's Avatar
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    Quote Originally Posted by xwrathbringerx View Post
    Oops sori

    S = sum of the lengths of the chords AC and BC
    ok, then only resitute "b" in ii)
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  5. #5
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    What exactly do you get for iii
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