In figure 3.78, 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

(1) Given RD = 15, DS = 4, MD = 8

MD × DN = RD × DS

This property is known as theorem of chords intersecting inside the circle.

⇒ 8 × DN = 15 × 4

(2)Given RS = 18, MD = 9, DN = 8

Here, RS = 18

Let RD = x and DS = 18 – x

MD × DN = RD × DS

This property is known as theorem of chords intersecting inside the circle.

⇒ 8 × 9 = x × (18 – x)

⇒18x – x² = 72

⇒x² – 18x + 72 = 0

⇒ (x – 12)(x – 6) = 0

⇒ x = 12 or 6

⇒ DS = 6 or 12