There is much more to learn about chords than their relationship to the diameter. Suppose you encounter a problem involving two chords that intersect each other within the same circle. Are there special rules that apply to such situations, and can you use these rules to solve a problem? Study the slides below to see how your knowledge of chords can help you find the length of a segment formed when two chords intersect.

In this circle, \(\small\mathsf{ \overline {GM}}\) and \(\small\mathsf{ \overline {KF}}\) intersect at point N. As a result, four segments are formed: \(\small\mathsf{ \overline {KN}}\) \(\small\mathsf{ \overline {FN}}\) \(\small\mathsf{ \overline {MN}}\) \(\small\mathsf{ \overline {GN}}\) If you're given the information that KN = 8, NM = 6, and NG = 10, how can you find the length of \(\small\mathsf{ \overline {NF}}\)? There's actually a theorem that can help you answer this question: Intersecting Chords Theorem: If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. How can you write this theorem in mathematical terms as an equation? KN(NF) = NG(NM) Now, see if you can solve the problem in your notebook and find NF. (Let NF be x in the equation.) 8x = 10(6) 8x = 60 x = 60/8 x = 7.5 Now, use the same procedure to find the value of x in this circle. 9x = 4(12) 9x = 48 \(\small\mathsf x { \approx {} }5.3\)