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1. In triangle  ABC, AB=BC=17 and AC=16 Find the circumradius of triangle ABC. 

- I have gotten two wrong answers which were both 15/2 and 17/2... Plz halp I do want an explantion just to undersand.

 

2. In triangle ABC, M is the midpoint of line AB Let D be the point on line BC such that line AD bisects angle BAC, and let the perpendicular bisector of  line AB intersect line AD at E If AB=44 and ME=12 then find the distance from E to line  AC. 

- I have seen similar questions like these posted but I personly cant understand those. 

 

THX in ADVANCE

 Apr 24, 2020
 #2
avatar+20798 
+1

1)  I'm going to put this on a coordinate axis.

     I place point A at the origin:  A = (0,0)

     I place point C at the point (16,0)

     Since this is an isosceles triangle, point B will be somewhere above the midpoint of AC.

     The distance from A to this midpoint is 8; the distance from A to point B is 17.

     This makes an 8 - 15 - 17 right triangle, so point B = (8,15)

     The circumcircle has its center at the intersection points of the perpendicular bisectors of the sides. 

     The line drawn from point B to the midpoint of AC is one of these perpendicular bisectors and its equation is x = 8.

     Now, I plan to find the perpendicular bisector of AB and find where this line crosses the line x = 8.

     The slope of AB is:  (15 - 0) / (8 - 0)  =  15/8.

     This means that the perpendicular bisector of AB has a slope of -8/15.

     The midpoint of AB is the point (4, 7.5).

     The equation of the perpendicular bisector of AB is:  y - 7.5  =  (-8/15)(x - 4)

      Multiplying this out:  15y - 112.5  =  -8x + 32     --->     8x + 15y  =  144.5.

      Now, to find where this line intersects the line x = 8:    8(8) + 15y  =  144.5   --->   15y  =  80.5

                     --->   y  =  5.03125

      This means that the circumcenter is (8, 5.03125)

       To find the radius, I will find the distance from this point to the point (0,0):

            distance  =  sqrt( (8.5 - 0)2 + (5.03125 - 0)2 )  =  sqrt( 97.56347656 )  =  9.877422567.

 

I hope that this is clear and I hope that I didn't mess it up!

 Apr 24, 2020
 #6
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+1

Check this solution: https://www.wolframalpha.com/input/?i=Triangle+with+sides+17%2C+17%2C+16%2C+calculate+the+circumradius

Guest Apr 25, 2020
 #3
avatar+20798 
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2)  If I understand the question correctly:

     There is an angle(BAC).

      AD bisectes this angle.

      E is on AD.

      M is a point on AB and ME is perpendicular to AB.

      Therefore, the length of ME is the distance from E to line segment AB.

      Since every point on the bisector of an angle is equidistant from the sides of the angle, the distance from E to AB 

           is the same as the distance from E to AC.

      Since distances are measured perpendicular to segments, E to AC must equal E to AB, so the distance from E to AC

           must also be 12.

 Apr 25, 2020
 #4
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1. I understand you reasoning and I thank you for it! I have a hard time understanding decimals tho.. THX I still understand and I will probly use it in the future

2. THX, I thought it was more complicated that that lol!

I might not fully understand 1 but I went from a 0 undersding to a 85 out of 100. Thanks for all of your help!

 Apr 25, 2020
 #5
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answer 1 is not right... Thx for trying tho... :D

 Apr 25, 2020
 #7
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Acute isosceles triangle.
Sides: a = 17 b = 17 c = 16

Area: T = 120
Perimeter: p = 50
Semiperimeter: s = 25

Angle ∠ A = α = 61.928° = 61°55'39″ = 1.081 rad
Angle ∠ B = β = 61.928° = 61°55'39″ = 1.081 rad
Angle ∠ C = γ = 56.145° = 56°8'42″ = 0.98 rad

Height: ha = 14.118
Height: hb = 14.118
Height: hc = 15

Median: ma = 14.151
Median: mb = 14.151
Median: mc = 15

Inradius: r = 4.8
Circumradius
The circumcircle of a triangle is a circle that passes through all of the triangle's vertices, and the circumradius of a triangle is the radius of the triangle's circumcircle. Circumcenter (center of circumcircle) is the point where the perpendicular bisectors of a triangle intersect.
R =abc / 4rs
Circumradius: R = 9.633

Guest Apr 25, 2020
 #8
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Thx but what is it in fraction form? I am confused on that portion of it :p

Guest Apr 25, 2020
 #9
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R=9  19/30

Guest Apr 25, 2020
 #10
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I wish... I still cant get the right answer... I wish I understood more... Thx for trying all of everyone on this thread. I might ask agian cuz I have gotten it wrong many times now lol. 

Guest Apr 25, 2020
 #11
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R= 289 / 30

Guest Apr 25, 2020
 #12
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Thank you!

Guest Apr 25, 2020
edited by Guest  Apr 25, 2020
edited by Guest  Apr 25, 2020

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