Difference between revisions of "2021 AMC 12B Problems/Problem 8"

Revision as of 18:47, 16 February 2021

Problem

Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38,38,$ and $34$ . What is the distance between two adjacent parallel lines?

$\textbf{(A) }5\frac12 \qquad \textbf{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textbf{(D) }7 \qquad \textbf{(E) }7\frac12$

Solution 1

$[asy] size(6cm); pair O = (0, 4), A = (0, 5), B = (0, 7), R = (3.873, 5), L = (2.645, 7); draw(O--A--B); draw(O--R); draw(O--L); label("$A$", A, NW); label("$B$", B, N); label("$R$", R, NE); label("$L$", L, N); label("$O$", O, S); label("$d$", O--A, W); label("$2d$", A--B, W*2+0.5*N); label("$r$", O--R, S); label("$r$", O--L, S*0.5 + 1.5 * E); dot(O); dot(A); dot(B); dot(R); dot(L); draw(circle((0, 4), 4)); draw((-3.873, 3) -- (3.873, 3)); draw((-3.873, 5) -- (3.873, 5)); draw((-2.645, 7) -- (2.645, 7)); [/asy]$

Since two parallel chords have the same length (38), they must be equidistant from the center of the circle. Let the perpendicular distance of each chord from the center of the circle be $d$ . Thus, the distance from the center of the circle to the chord of length $34$ is

$2d + d = 3d$

and the distance between each of the chords is just $2d$ . Let the radius of the circle be $r$ . Drawing radii to the points where the lines intersect the circle, we create two different right triangles:

- One with base $\frac{38}{2}= 19$ , height $d$ , and hypotenuse $r$ ( $\triangle RAO$ on the diagram)

- Another with base $\frac{34}{2} = 17$ , height $3d$ , and hypotenuse $r$ ( $\triangle LBO$ on the diagram)

By the Pythagorean theorem, we can create the following system of equations:

$19^2 + d^2 = r^2$

$17^2 + (3d)^2 = r^2$

Solving, we find $d = 3$ , so $2d = \boxed{\textbf{(B)}\ 6}$ .

-Solution by Joeya and diagram by Jamess2022(burntTacos). (Someone fix the diagram if possible. - Done. )

Solution 2 (Coordinates)

Because we know that the equation of a circle is $(x-a)^2 + (y-b)^2 = r^2$ where the center of the circle is $(a, b)$ and the radius is $r$ , we can find the equation of this circle by centering it on the origin. Doing this, we get that the equation is $x^2 + y^2 = r^2$ . Now, we can set the distance between the chords as $2d$ so the distance from the chord with length 38 to the diameter is $d$ .

Therefore, the following points are on the circle as the y-axis splits the chord in half, that is where we get our x value:

$(19, d)$

$(19, -d)$

$(17, -3d)$

Now, we can plug one of the first two value in as well as the last one to get the following equations:

$19^2 + d^2 = r^2$

$17^2 + (3d)^2 = r^2$

Subtracting these two equations, we get $19^2 - 17^2 = 8d^2$ - therefore, we get $72 = 8d^2 \rightarrow d^2 = 9 \rightarrow d = 3$ . We want to find $2d = 6$ because that's the distance between two chords. So, our answer is $\boxed{B}$ .

~Tony_Li2007

Video Solution by Hawk Math

https://www.youtube.com/watch?v=VzwxbsuSQ80

Video Solution by Punxsutawney Phil

https://youtu.be/yxt8-rUUosI

Video Solution by OmegaLearn (Circular Geometry)

https://youtu.be/XNYq4ZMBtBU

Revision as of 20:35, 13 February 2021 (view source) Joeya (talk \| contribs) m (→‎Solution 1) ← Older edit		Revision as of 18:47, 16 February 2021 (view source) Biketoday (talk \| contribs) (→‎Solution 1) Newer edit →
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−	Since ~~the~~ two chords ~~of length <math>38</math>~~ 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 <math>d</math>. Thus, the distance from the center of the circle to the chord of length <math>34</math> is	+	Since two parallel chords have the same length (38), they must be equidistant from the center of the circle. Let the perpendicular distance of each chord from the center of the circle be <math>d</math>. Thus, the distance from the center of the circle to the chord of length <math>34</math> is

	<cmath>2d + d = 3d</cmath>		<cmath>2d + d = 3d</cmath>

2021 AMC 12B (Problems • Answer Key • Resources)
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