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Inradius = [ ABC ] / [semiperimeter of ABC]

ABC is a right triagle so  the area =  (1/2)(product of the legs)  = (1/2) (5 * 12) =  30

Semi-perimeter  =  [ 5 + 12 + 13 ] / 2  =  15

Inradius =  (30) / ( 15)  = 2

Because ID is parallel to AC, then AE = AC - ID = 12 - 2  = 10

Draw AI.....this  bisects angle A

And  IE = 2

And triangle AIE is right because angle AEI  = 90°

So....the area of AEI = (1/2)(2)(10) =  10

And by   SAS, triangle  AFI is congruent to triangle AEI....so its area also = 10

So  [ AEIF ]  =   10+  10  =   20

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"!

25^(1/2) - log ₅ sqrt (5)  = 

sqrt (25)  - log ₅  5^(1/2)  =

5  -  (1/2) log ₅ 5          {  log ₅ 5  =  1 }

5 - (1/2)(1)  =

5 - 1/2 = 

9/2

Nice job, 3Mathketeers !!!!!

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 

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.

The difference of the  roots = 

 sqrt ( 30^2  - 4 * 180) =   sqrt [ 900 - 720 ] =   sqrt [ 180]   =   sqrt [ 36 * 5 ]  =  6 sqrt (5) 

Sum =  1 + 7/8 =   15 / 8

The sequence is not geometric. 

I think it is 2. 

Thank you so much CPhill :))

Inverse of a 2 x 2 matrix =

1/ (determinant)  [ d  -b ]

                            [ -c  a]

det  = (2 *-4)  - (-1 * 6) =   -8 + 6 =   -2

So  the inverse matrix  =     (-1/2) [- 4   1  ]  =    [  2    -1/2 ]    

                                                     [  -6    2 ]        [   3    -1 ]      

First blank  "equal"

Second blank "are"

Another ChatGPT generated message. *sigh* Bro, ur profile image is literally CPhiII having a sad face. 

Please don't be rude to Cphill. Being rude to Cphill is not ok, even if you disagree with them or think they are wrong. Instead of downvoting peoples posts, respect is important, and we should all treat each other with kindness. Let’s leave this in the past, and move on with the conversation.

I think it is 2. Not sure. 

CPhill, this answer is ChatGPT-generated. Even though the answer is correct, please be more careful around such answers. 

tastyabananas2ndDad is not tastybanana, CPhiII. At least, I do not think it is. 

Nice job, tastybananas !!!