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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?

 

 Aug 14, 2023

Best Answer 

 #1
avatar+177 
+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.

 Aug 14, 2023
edited by The3Mathketeers  Aug 14, 2023
 #1
avatar+177 
+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
edited by The3Mathketeers  Aug 14, 2023
 #3
avatar+126976 
0

Nice job, 3Mathketeers !!!!!

 

 

cool cool cool

CPhill  Aug 14, 2023
 #4
avatar+177 
+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
 #5
avatar+126976 
+1

LOL!!!!......can't argue with that !!!!

 

 

cool cool cool

CPhill  Aug 14, 2023
 #2
avatar+126976 
+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 

 

 

cool cool cool

 Aug 14, 2023

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