Question

Two parallel chords of a circle are on the opposite sides of the centre of a circle of radius 25 cm. If one chord is 48 cm long, find the length of the other chord if the distance between the chords is 27 cm.

Answer 1

Given:

• Radius of the circle r = 25 cm,
• The length of the first chord l₁ = 48 cm,
• The distance between the two parallel chords d = 27 cm.

Let the length of the second chord be l₂ and let the distance from the center of the circle to the first chord be d₁.

Let the distance from the center of the circle to the second chord be d₂.

Since the chords lie on opposite sides of the center, the distance between them is the sum of the distances of the chords from the center:

d₁ + d₂ = 27 cm

So, we can write:

d₂ = 27 – d₁

Step 1: For the first chord (length l₁ = 48 cm):

Half of the length of the first chord is:

`48/2 = 24` 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, we get:

`r^2 = d_1^2 + (l_1/2)^2`

Substitute the known values:

`25^2 = d_1^2 + 24^2`

`625 = d_1^2 + 576`

`d_1^2 = 625 - 576 = 49`

`d_1 = sqrt(49) = 7` cm

Step 2: For the second chord (length l₂):

Now, for the second chord, let l₂ be the length of the second chord.

Half of the length of the second chord is `l_2/2`​​. 

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

`r^2 = d_2^2 + (l_2/2)^2`

Substitute the known values:

`25^2 = (27 - 7)^2 + (l_2/2)^2`

`625 = 20^2 + (l_2/2)^2`

`625 = 400 + (l_2/2)^2`

`(l_2/2)^2 = 625 - 400 = 225`

`l_2/2 = sqrt(225) = 15`

l₂ = 2 × 15 = 30 cm

The length of the second chord is 30 cm.