¶ Spring 2021 AMC 12B Problem 8
Problem:
Three equally spaced parallel lines intersect a circle, creating three chords of lengths , , and . What is the distance between two adjacent parallel lines?
Answer Choices:
A.
B.
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E.
Solution:
Since two parallel chords have the same length , they must be equidistant from the center of the circle. Let the perpendicular distance of each chord from the center of the circle be . Thus, the distance from the center of the circle to the chord of length is and the distance between each of the chords is just .
Let the radius of the circle be . Drawing radii to the points where the lines intersect the circle, we create two different right triangles:
By the Pythagorean theorem, we can create the following system of equations:
Subtracting these two equations, we get - therefore, we get . We want to find because that's the distance between two chords. So, our answer is
Because we know that the equation of a circle is where the center of the circle is and the radius is , we can find the equation of this circle by centering it on the origin. Doing this, we get that the equation is . Now, we can set the distance between the chords as so the distance from the chord with length 38 to the diameter is .
Therefore, the following points are on the circle as the -axis splits the chord in half, that is where we get our value:
Now, we can plug one of the first two value in as well as the last one to get the following equations:
Solving, we find , so
The problems on this page are the property of the MAA's American Mathematics Competitions