Two concentric circles have radii 8 and 17, respectively. What is the length of the chord in the larger circle created by a tangent line to the smaller circle?

sdfdsasdfafdsaf Aug 14, 2023

#1**+1 **

I have created a diagram with labeled points so that you can follow along:

In order to solve this problem and find the length \(\rm BC\), we will take advantage of the Chord Intersecting Theorem. This theorem states that, for two intersecting chords within a circle, \({\rm DA} * {\rm AE} = {\rm BA} * {\rm AC}\). In addition, this problem states that \(\overline{\rm BC}\) is tangent to the smaller circle, which means that this segment is perpendicular to the radius of the smaller circle. Since the radius of the larger circle intersects with the chord of the larger circle at a right angle, \(\overline{\rm OD}\) is also a perpendicular bisector of the chord. I have marked this in the diagram. Now, just apply the Chord Intersecting Theorem and solve for for the unknown:

\({\rm DA} * {\rm AE} = {\rm BA} * {\rm AC} \\ 9 * (8 + 8 + 9) = \frac{1}{2}l * \frac{1}{2}l \\ 9 * 25 = \frac{1}{4}l^2 \\ l^2 = 900 \\ l = 30 \text{ or } l = -30\)

Since we are dealing with distances, reject \(l = -30\). Therefore, L = 30.

The3Mathketeers Aug 14, 2023

#1**+1 **

Best Answer

I have created a diagram with labeled points so that you can follow along:

In order to solve this problem and find the length \(\rm BC\), we will take advantage of the Chord Intersecting Theorem. This theorem states that, for two intersecting chords within a circle, \({\rm DA} * {\rm AE} = {\rm BA} * {\rm AC}\). In addition, this problem states that \(\overline{\rm BC}\) is tangent to the smaller circle, which means that this segment is perpendicular to the radius of the smaller circle. Since the radius of the larger circle intersects with the chord of the larger circle at a right angle, \(\overline{\rm OD}\) is also a perpendicular bisector of the chord. I have marked this in the diagram. Now, just apply the Chord Intersecting Theorem and solve for for the unknown:

\({\rm DA} * {\rm AE} = {\rm BA} * {\rm AC} \\ 9 * (8 + 8 + 9) = \frac{1}{2}l * \frac{1}{2}l \\ 9 * 25 = \frac{1}{4}l^2 \\ l^2 = 900 \\ l = 30 \text{ or } l = -30\)

Since we are dealing with distances, reject \(l = -30\). Therefore, L = 30.

The3Mathketeers Aug 14, 2023

#4**+1 **

Thanks for the compliment! Your method is equally as valid and potentially quicker than mine. I can claim to have bragging rights as I have been officially featured as "Best Answer"!

The3Mathketeers
Aug 14, 2023

#2**+1 **

From the center of the circle, draw radius 8 to the midpoint of the chord

Then from the center of the circle draw radius 17 to either endpoint of the chord

This will will create a right triangle with a hypotenuse of 17 and one leg = 8

Half of the chord length = the other leg of the right triangle= sqrt [ 17^2 - 8^2 ] = sqrt (225) = 15

So.....the length of the chord = 2 (15) = 30

CPhill Aug 14, 2023