Two parallel chords of lengths 6 cm and 8 cm are on opposite sides of the centre of a circle. Class 9

Two parallel chords of lengths 6 cm and 8 cm are on opposite sides of the centre of a circle. Class 9

Question 1.

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. Class 9

Solution:

Let AB = 6 cm and CD = 8 cm be two parallel chords on opposite sides of the centre O, with midpoints M and N respectively.

Join OA, OC, OM and ON.

OA = OC = 5 cm ...[Radii of the same circle]

Since the perpendicular from the centre of a circle to a chord bisects the chord.

∴ AM = $\frac{\mathrm{AB}}{2}=\frac{6}{2}$ = 3 cm

∴ CN= $\frac{\mathrm{CD}}{2}=\frac{8}{2}$ = 4 cm

In right-angled ∆OMA,

OA² = OM² + AM²

...[By Baudhāyana-Pythagoras theorem]

∴ 5² = OM² + 3²

∴ 25 = OM² + 9

∴ OM² = 16

∴ OM = 4 cm

Similarly, in right-angled ∆ONC,

OC² = ON² + CN²

...[By Baudhāyana-Pythagoras theorem]

∴ 5² = ON² + 4²

∴ 25 = ON² + 16

∴ ON² = 9

∴ ON = 3 cm

Since the chords are on opposite sides of the centre,

MN = OM + ON = 4 + 3 = 7 cm

∴ Distance between the midpoints of the chords = 7 cm