Question

Two parallel chords in a circle are 10 cm and 24 cm long. If the radius of the circle is 13 cm, find the distance between the chords if they lie:

1. on the same side of the centre
2. on the opposite sides of the centre.

Answer 1

Given:

• Radius of the circle r = 13 cm,
• Length of the first chord = 10 cm,
• Length of the second chord = 24 cm.

Let the distance of the first chord from the center of the circle be d₁ and the distance of the second chord from the center be d₂.

1. For the chord of length 10 cm:

Let the distance from the center to the chord of length 10 cm be d₁.

Half the length of this chord is:

`10/2 = 5` cm

Using the Pythagorean theorem for the right-angled triangle formed by the radius, the distance from the center to the chord and half the length of the chord:

`r^2 = d_1^2 + (10/2)^2`

Substitute the values:

`13^2 = d_1^2 + 5^2`

`169 = d_1^2 + 25`

`d_1^2 = 169 - 25 = 144`

`d_1 = sqrt(144) = 12` cm

2. For the chord of length 24 cm:

Let the distance from the center to the chord of length 24 cm be d₂​.

Half the length of this chord is:

`24/2 = 12` cm

Again, using the Pythagorean theorem for the right-angled triangle formed by the radius, the distance from the center to the chord and half the length of the chord:

`r^2 = d_2^2 + (24/2)^2`

Substitute the values:

`13^2 = d_2^2 + 12^2`

`169 = d_2^2 + 144`

`d_2^2 = 169 - 144 = 25`

`d_2 = sqrt(25) = 5` cm

Now, let’s calculate the distance between the two chords.

Case 1: Chords on the same side of the center

If the two chords lie on the same side of the center, the distance between them is:

Distance between chords

= |d₁ – d₂|

= |12 – 5|

= 7 cm

Case 2: Chords on opposite sides of the center

If the two chords lie on opposite sides of the center, the distance between them is:

Distance between chords

= d₁ + d₂

= 12 + 5

= 17 cm