Two parallel chords of lengths 6 cm and 8 cm are on opposite sides of the centre of a circle. If the radius of the circle is 5 cm, find the distance between the midpoints of the chords.                                                         [Page No. 101]

Let O be the centre of the circle.

Length of the two chords are AB = 6 cm and CD = 8 cm.

Given: Radius of the circle OA = OC = 5 cm

Let us draw OM ⊥ AB and ON ⊥ CD.

The perpendiculars from the centre O to the chords AB and CD bisects the chords at M and N respectively.

For chord CD = 8 cm: CN = 4 cm.

Using Pythagoras in ΔOCN:

$$ON^2 = OC^2 - CN^2 = 5^2 - 4^2 = 25 - 16 = 9$$

So, ON = 3 cm.

For chord AB = 6 cm: AM = 3 cm.

Using Pythagoras in ΔOAM:

$$OM^2 = OA^2 - AM^2 = 5^2 - 3^2 = 25 - 9 = 16$$

So, OM = 4 cm.

Since, the chords are on opposite sides of the centre and their centres lies on same line.

So, the distance between the midpoints

$$= ON + OM = 3 + 4 = 7 \text{ cm.}$$