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Let triangle ABC be an equilateral triangle. There is a point O insie the triangle such that angle AOB = 115 degrees and angle BOC = 125 degrees. Ir a triangle is constructed with sides with length AO, BO, and CO, find the degree measure of the largest interior angle of this triangle.

 May 17, 2021
 #1
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I have not been able to work it out algebraically.

 

So I estimated it graphically. (using a free program GeoGebra)    [Not what you want, I know this]

 

The stupid pic has deleted itself so I cannot show it to you.  bummer!  

(Geogebra is playing up - this has not happened to me before)

 

 

 

The sides are approximately  in the ratio  65:62:59

The biggest angle is opposite the longest side.

Using cosine rule I got the biggest angle as approx 64.9 degrees.

 

 

Anyway. I am sure there is some  sensible algebraic way to do it.   indecision

 May 20, 2021
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Turns out that the largest angle is 66.468 (3 dp ) deg.

Call the angle OAC alpha say, and run that round the triangle, 

OAB = 60 - alpha, OBA = 180 - 115 - (60 - alpha), and so on.

Calling AO x, BO y and CO  z, then use the sin rule twice, in triangles AOB and AOC,

equate the two to derive an expression for tan (alpha) - I got alpha = 28.3296 deg.

From that, using the sin rule in the three triangles, calculate x, y an z.

I got (to 4 dp) x = 0.6063, y = 0.5793, z = 0.5236.

Then the cosine rule to calculate the largest angle.

 May 20, 2021

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