Question

Two parallel chords of length 4.4cm and 9cm are in a circle of radius 7.5cm. Calculate the distance between the two chords if they are on the same side of the circle

Answer 1

To find the distance between the two parallel chords in a circle, we can use the relationship between the radius of the circle, the length of the chords, and their distances from the center of the circle.

Let's denote:
- \( r = 7.5 \) cm (radius of the circle)
- \( L_1 = 4.4 \) cm (length of the first chord)
- \( L_2 = 9 \) cm (length of the second chord)
- \( d_1 \) = distance from the center of the circle to the first chord
- \( d_2 \) = distance from the center of the circle to the second chord

We can use the following relationship for each chord:
\[
d^2 + \left(\frac{L}{2}\right)^2 = r^2
\]
where \( d \) is the distance from the center of the circle to the chord and \( L \) is the length of the chord.

First, let's calculate \( d_1 \) for the first chord (length 4.4 cm):
\[
\frac{L_1}{2} = \frac{4.4}{2} = 2.2 \text{ cm}
\]
Using the relationship:
\[
d_1^2 + (2.2)^2 = (7.5)^2
\]
Calculating \( (2.2)^2 \):
\[
(2.2)^2 = 4.84
\]
Calculating \( (7.5)^2 \):
\[
(7.5)^2 = 56.25
\]
Now substituting these values into the equation:
\[
d_1^2 + 4.84 = 56.25
\]
\[
d_1^2 = 56.25 - 4.84
\]
\[
d_1^2 = 51.41
\]
\[
d_1 = \sqrt{51.41} \approx 7.17 \text{ cm}
\]

Next, let's calculate \( d_2 \) for the second chord (length 9 cm):
\[
\frac{L_2}{2} = \frac{9}{2} = 4.5 \text{ cm}
\]
Using the same relationship:
\[
d_2^2 + (4.5)^2 = (7.5)^2
\]
Calculating \( (4.5)^2 \):
\[
(4.5)^2 = 20.25
\]
Now substituting into the equation:
\[
d_2^2 + 20.25 = 56.25
\]
\[
d_2^2 = 56.25 - 20.25
\]
\[
d_2^2 = 36
\]
\[
d_2 = \sqrt{36} = 6 \text{ cm}
\]

Finally, we need to find the distance between the two chords. Since the chords are on the same side of the circle, the distance between them is:
\[
\text{Distance} = |d_1 - d_2| = |7.17 - 6| = 1.17 \text{ cm}
\]

Thus, the distance between the two chords is approximately **1.17 cm**.