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1. In right triangle \(ABC\) \(,\angle C = 90^\circ. \)Midian \(\overline{AM} \) has a length of 19, and a median \(\overline{BN}\) has a length of 13. What is the length of the hypotenuse of the triangle? The answer has a square root.

 

 

2. A circle has a radius of 14. Let \(\overline {AB}\) be a chord of the circle, such that AB=12. What is the distance between the chord and the center of the circle? The answer will be a square root.

 

 

If you can give me the answer AND explain how you got it that would be great! I want to understand not just get the answers :/ 

Thanks in advance!

-Wolf

 Apr 19, 2020
 #1
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Number 1:

 

Point M divides BC into two equal segments; call each segment "x".

Point N divides AC into two equal segments; call each segment "y".

 

Triangle(BCN) is a right triangle with BC = 2x, CN = y, and hypotenuse BN = 13.

--->   (2x)2 + (y)2  =  (13)2   --->   4x2 + y2  =  169          [Pythagorean Theorem]

 

Triangle(AMC) is a right triangle, with AC = 2y, CM = x, and hypotenuse AM = 19.

--->   (2y)2 + (x)2  =  (19)2   --->   4y2 + x2  =  361          [Pythagorean Theorem]

 

Combining these two equations:

     4x2 +   y2  =  169   --->   x 4    --->   16x2 + 4y2  =   676

       x2 + 4y2  =  361   --->   x -1   --->      -x2 - 4y2  =  -361

Adding down:                                        15x2           =   315

                                                                          x2   =  21

                                                                           x   =  sqrt(21)

 

Substituting:   4x2 +   y2  =  169   --->   4( sqrt(21) )2 + y2  =  169

                                                                           84    + y2  =  169

                                                                                      y2  =  85

                                                                                      y   =  sqrt(85)

 

This means that one side of triangle(ABC)  =  2·sqrt(21)  while the other side  =  2·sqrt(85).

 

You can now use the Pythagorean Theorem to find the hypotenuse ...

 Apr 19, 2020
 #2
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thx I understand this question now :D 

Guest Apr 19, 2020
 #3
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Number 2)  The circle has radius = 14, so its diameter = 28. The chord = 12

 

Draw the chord and the diameter of the circle which is perpendicular to the chord.

 

The chord divides the diameter into two parts, a short part from the chord to the circle (call this "x") 

and a longer part from the chord through the center of the circle to the other end of the circle (this is 28 - x).

 

The chord is divided into two equal parts, each part = 6.

 

There is a theorem that says that if you multiply the two parts of one chord you will get the same answer when

you multiply the two parts of the other chord (the diameter).  Thus:  x·(28 - x)  =  6·6

 

Solving:  28x - x2  =  36

               x2 - 28x + 36  =  0

Using the quadratic formula:  x  =  1,351  (approximately)

 

The distance from the chord to the center of the circle is 14 - 1.351  =  12.649   (approximately)

 

After doing this, I realize that there is a much easier way --

-- draw the chord and the perpendicular bisector (the same as above)

-- draw the radius from the center of the circle to an end of the chord, creating a right triangle.

-- the hypotenuse of the triangle is the radius of the circle (14)

-- one leg of the right triangle is one-half the chord (6)

-- use the Pythagorean Theorem to find the distance ...

 Apr 19, 2020
 #4
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Thank you so much! this helps me but I am still confused how to convert that answer to a squre root? If you could help with that I would be super gratfull! Thx I understand how to do it know I just am now confused with the square root :P

Guest Apr 19, 2020
 #5
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I cant edit my prevous comment because of the whole guest thing, but what I ment to say is where are the legs of the triangle.. Is it a right triangle cuz then i can just use the pythagrom therum? (I cant spell well) If so what is the other leg? I understand that one of the legs is 6 but what about the other? would it be 6 as well? Thx so much

Guest Apr 19, 2020
 #6
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Thanks so much! I got the answer after re-reading your explanation! 

 Apr 19, 2020

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