# Circle Geometry

• October 7th 2008, 12:52 AM
xwrathbringerx
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?

Thanx very much
• October 7th 2008, 05:39 PM
Nacho
i) because <a always takes the same arc AB

ii) using "sin theorem"

$
\frac{{\sin a}}
{K} = \frac{{\sin b}}
{{AC}} \Leftrightarrow AC = \frac{{K\sin b}}
{{\sin a}}
$

again

$
\frac{{\sin a}}
{K} = \frac{{\sin \left( {180 - (a + b)} \right)}}
{{BC}} \Leftrightarrow \frac{{\sin a}}
{K} = \frac{{\sin \left( {a + b} \right)}}
{{BC}} \Leftrightarrow BC = \frac{{K\sin \left( {a + b} \right)}}
{{\sin a}}
$

suming

$
AC + BC = \frac{{K\sin b}}
{{\sin a}} + \frac{{K\sin \left( {a + b} \right)}}
{{\sin a}} = \frac{K}
{{\sin a}}\left( {\sin b + \sin \left( {a + b} \right)} \right)
$

What is S?
• October 7th 2008, 05:45 PM
xwrathbringerx
Oops sori

S = sum of the lengths of the chords AC and BC
• October 7th 2008, 07:03 PM
Nacho
Quote:

Originally Posted by xwrathbringerx
Oops sori

S = sum of the lengths of the chords AC and BC

ok, then only resitute "b" in ii)
• October 11th 2008, 03:21 AM
xwrathbringerx
What exactly do you get for iii