Sorry to hear of your computer troubles. I solved this a bit faster with trig functions, although I do normally like the challenge of not using trig. Let x be the angle complementary to your purple angle (i.e., the bottom angle made by the right side of the square). We can determine this angle from the side lengths of the right triangle that it's in. tan(x) = 5/8 x = arctan(5/8) Let y be the angle supplementary to the purple angle, which is just x plus 90 degrees. This angle is also the angle between the two tangent segments in the diagram with length 1. y = x + (pi/2) The tangent segment of length 1 and the radius make a right triangle, and the angle opposite the radius is half of y. We can therefore use trig functions again to define this relationship. tan(y/2) = R/1 = R R = tan([arctan(5/8) + (pi/2)]/2) ~= 1.8
The disadvantage of using the methods of trig functions is that your answer is coming in terms of trig functions. This would be fine if we just wanted the numerical answer but sometimes, we want it in the form an algebraic expression. If you can find a way to simplify the expression, that would be great.
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Thank you! :)
Sorry to hear of your computer troubles. I solved this a bit faster with trig functions, although I do normally like the challenge of not using trig.
Let x be the angle complementary to your purple angle (i.e., the bottom angle made by the right side of the square). We can determine this angle from the side lengths of the right triangle that it's in.
tan(x) = 5/8
x = arctan(5/8)
Let y be the angle supplementary to the purple angle, which is just x plus 90 degrees. This angle is also the angle between the two tangent segments in the diagram with length 1.
y = x + (pi/2)
The tangent segment of length 1 and the radius make a right triangle, and the angle opposite the radius is half of y. We can therefore use trig functions again to define this relationship.
tan(y/2) = R/1 = R
R = tan([arctan(5/8) + (pi/2)]/2) ~= 1.8
The disadvantage of using the methods of trig functions is that your answer is coming in terms of trig functions. This would be fine if we just wanted the numerical answer but sometimes, we want it in the form an algebraic expression. If you can find a way to simplify the expression, that would be great.