Solve the two questions given in the image.
Understand the Problem
The image presents two geometry problems. The first involves finding the distance between two parallel and equal chords in a circle. The second involves using a given line AB to answer further questions.
Answer
$2\sqrt{133}$ cm
Answer for screen readers
$2\sqrt{133}$ cm
Steps to Solve
- Find the distance from the center to each chord
Draw a perpendicular line from the center of the circle to each chord. This line bisects the chord. We can form a right triangle with the radius of the circle as the hypotenuse, half the chord length as one leg, and the distance from the center to the chord as the other leg. Let $d$ be the distance from the center to each chord. Using the Pythagorean theorem:
$d^2 + (12/2)^2 = 13^2$
$d^2 + 6^2 = 13^2$
$d^2 + 36 = 169$
$d^2 = 169 - 36$
$d^2 = 133$
$d = \sqrt{133}$
- Determine the position of the chords
Since the chords are parallel and equal in length, they can be on the same side of the center or on opposite sides.
- Calculate the distance between the chords when they are on opposite sides of the center
In this case, the distance between the chords is the sum of the distances from the center to each chord, which is $2d = 2\sqrt{133}$.
- Calculate the distance between the chords when they are on the same side of the center
In this case, the distance between the chords would be $0$, since this could only happen if the two chords were the same chord. Therefore, this case would not result in two different chords.
- Distance between the chords
Assuming the chords are on opposite sides of the center: The distance is $2\sqrt{133}$ cm.
$2\sqrt{133}$ cm
More Information
The value of $2\sqrt{133}$ is approximately 23.065 cm.
Tips
A common mistake is to not consider the two possible arrangements of the chords relative to the center of the circle (same side or opposite sides). In this problem we are not given any more data to determine whether the two chords are on the same side or opposite sides of the center. If they are on opposite sides, the distance is $2\sqrt{133}$, while if they are on the same side, the distance has to be zero because they are said to be two chords, and would not be distinguishable otherwise.
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