Two parallel chords of length 5 units and 8 units are on opposite sides of the center of a circle of radius 7 units. What is the distance between the chords? (round your answer to two decimal places)

The correct answer is 12.28 units.

Step-by-step Explanation:

Let us analyze the given information about the circle and the two parallel chords:

1. Radius of the circle, $R=7$ units.
2. Length of the first chord, $L_1=5$ units.
3. Length of the second chord, $L_2=8$ units.
4. The two chords are situated on opposite sides of the center of the circle.

A fundamental property of circles states that a perpendicular line drawn from the center of a circle to a chord bisects the chord. Therefore, the perpendicular distance from the center to each chord can be calculated using the Pythagorean theorem in the right-angled triangle formed by the radius, half the chord length, and the perpendicular distance.

Step 1: Find the distance from the center to the first chord ($d_1$)

Half the length of the first chord is:

$\frac{L_1}{2}=\frac{5}{2}=2.5$

Using the Pythagorean theorem:

$d_1^2+{\left(2.5\right)}^2=R^2$

$d_1^2+6.25=7^2=49$

$d_1^2=49-6.25=42.75$

$d_1=\sqrt{42.75}\approx 6.5383$

Step 2: Find the distance from the center to the second chord ($d_2$)

Half the length of the second chord is:

$\frac{L_2}{2}=\frac{8}{2}=4$

Using the Pythagorean theorem:

$d_2^2+4^2=R^2$

$d_2^2+16=7^2=49$

$d_2^2=49-16=33$

$d_2=\sqrt{33}\approx 5.7446$

Step 3: Calculate the total distance between the two parallel chords

Since the chords lie on opposite sides of the center, the total distance $D$ between them is the sum of their individual perpendicular distances from the center:

$D=d_1+d_2$

$D=6.5383+5.7446=12.2829$

Rounding to two decimal places gives $12.28$ units.

Thus, the distance between the two chords is 12.28 units.