Problem 12

Point P is 9 units from the center of a circle of radius 15.

How many different chords of the circle contain P and have integer lengths?

(15-9) * (15+9)  =   6 *  24  = 144

$\begin{array}{|rcll|} \hline 144 &=& 2^4\cdot 3^2 \\ \hline \end{array}$

$\begin{array}{|rcll|} \hline 144 &=& (2\cdot 3) \cdot ( 2^3\cdot 3 ) \\ &=& 6 \cdot 24 \\\\ 144 &=& (2^3) \cdot ( 2\cdot 3^2 ) \\ &=& 8 \cdot 18 \\\\ 144 &=& (3^2) \cdot ( 2^4 ) \\ &=& 9 \cdot 16 \\\\ 144 &=& (2^2\cdot 3) \cdot (2^2\cdot 3) \\ &=& 12 \cdot 12 \\ \hline \end{array}$

1. chord ( length = 30 = 6+24 ) 

2. chord ( length = 26 = 8+18 )

3. chord ( length = 26 = 18+8 )

4. chord ( length = 25 = 9+16 )

5. chord ( length = 25 = 16+9 )

6. chord ( length = 24 = 12+12 )

see chord theorem: