Question

In the adjoining figure, chord MN and chord RS intersect at point D. 

i. If RD = 15, DS = 4, MD = 8 find DN 

ii. If RS = 18, MD = 9, DN = 8 find DS

Answer 1

i. Chords MN and RS intersect internally at point D. [Given] 

∴ MD × DN = RD × DS [Theorem of internal division of chords] 

∴ 8 × DN = 15 × 4 

∴ DN = (15 x 4)/8

∴ DN = 7.5 units 

ii. Let the value of RD be x. 

RS = RD + DS [R – D – S] 

∴ 18 = x + DS 

∴ DS = 18 – x

Now, MD × DN = RD × DS [Theorem of internal division of chords] 

∴ 9 × 8 = x(18 – x) 

∴ 72 = 18x – x² 

∴ x² – 18x + 72 = 0 

∴ x² – 12x – 6x + 72 = 0 

∴ x (x – 12) – 6 (x – 12) = 0 

∴ (x – 12) (x – 6) = 0 

∴ x – 12 = 0 or x – 6 = 0 

∴ x = 12 or x = 6 

∴ DS = 18 – 12 or DS = 18 – 6 

∴ DS = 6 units or DS = 12 units