SOLUTION: an isosceles triangle, whose legs are each 13, is inscribed in a circle. if the altitude to the base of the triangle is 5, find the radius of the circle.

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Question 388780: an isosceles triangle, whose legs are each 13, is inscribed in a circle. if the altitude to the base of the triangle is 5, find the radius of the circle.
Found 3 solutions by scott8148, Edwin McCravy, robertb:
Answer by scott8148(6628) About Me  (Show Source):
You can put this solution on YOUR website!
the altitude is on a diameter of the circle

5:13::13:d

5d = 169 ___ 10r = 169 ___ r = 16.9

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

The other tutor's solution is incorrect.



Draw in the radii to the vertices:

 

Extend the vertical radius down to the base of the given triangle,
which we label D, and we let x be the measure of OD
dividing the base into two equal parts, each measuring 2.5:



From right triangle BOD  

BD%5E2%2BDO%5E2=BO%5E2

r%5E2=2.5%5E2%2Bx%5E2

r%5E2=6.25%2Bx%5E2

r%5E2-x%5E2=6.25

%28r-x%29%28r%2Bx%29=6.25

From right triangle BAD

%28r%2Bx%29%5E2+%2B+2.5%5E2+=+13%5E2

%28r%2Bx%29%5E2+%2B+6.25+=+169

%28r%2Bx%29%5E2+%2B+6.25+=+169

%28r%2Bx%29%5E2+=+162.75

r%2Bx+=+%22%22+%2B-+sqrt%28162.75%29

Ignore the negative:

r%2Bx+=++sqrt%28162.75%29

r%2Bx+=++sqrt%28162.75%29

r%2Bx+=+12.75735082


Substitute sqrt%28162.75%29 for r%2Bx%29 in

%28r-x%29%28r%2Bx%29=6.25

%28r-x%29sqrt%28162.75%29=6.25

r-x=6.25%2Fsqrt%28162.75%29

r-x=0.489913626

Add these two equations:

system%7Br%2Bx+=+12.75735082%2C+r-x=0.489913626

2r = 13.24728445

 r = 6.62363632224

Edwin

Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Consider the isosceles triangle ABC, with A the vertex angle.Then AB = AC = 13. If we let D be the midpoint of segment BC, then segment AD is the altitude of triangle ABC from A to segment BC, and AD = 5, as given. This altitude AD must lie on a diameter, due to symmetry. Let AE be the diameter that contains the altitude AD. Then triangle ACE (in that order) is a right triangle, with C as the right angle.By similarity of the right triangle ACE with the right triangle ADC, we must have, by proportionality of corresponding sides, that d%2F13+=+13%2F5, or d = 169/5, or d = 33.8, where d = diameter of the circle. Thus r = 16.9.

Hope this helps!