Question

In the given figure, chord MN and chord RS intersect at point D.
(1) If RD = 15, DS = 4, MD = 8 find DN
(2) If RS = 18, MD = 9, DN = 8 find DS

Answer 1

If two chords of a circle intersect each other in the interior of the circle, then the product of the lengths of the two segments of one chord is equal to the product of the lengths of the two segments of the other chord.

(1) MD × DN = RD × DS
⇒ 8 × DN = 15 × 4
⇒ DN = \[\frac{60}{8}\] = 7.5 units

(2) MD × DN = RD × DS         ...[Theorem of internal division of chords] 
⇒ MD × DN = (RS − DS) × DS
⇒ 9 × 8 = (18 − DS) × DS
⇒ DS² − 18DS + 72 = 0
⇒ DS² − 12DS − 6DS + 72 = 0
⇒ DS(DS − 12) − 6(DS − 12) = 0
⇒ (DS − 12)(DS − 6) = 0
⇒ DS − 12 = 0 or DS − 6 = 0
⇒ DS = 12 units or DS = 6 units