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This is a high school level geometry question.

As shown in the top half of the image below, suppose we start with a right scalene triangle with hypotenuse = 5 and sides b and c = to 3 and 4, respectively.

Suppose we begin rotating the hypotenuse outward (rotated hypotenuse shown as dashed line "d" in bottom half of image). We rotate hypotenuse line "d" out by a unit of 1 from angle "B", denoted as dashed line "e" in the image. How do we measure lines f and g, formed in the new triangle as line d rotates outward?

It's obvious to me the new angle D formed as line "d" rotates will <> angle B. The little black square in the new triangle formed is my recollection of a 90 degree angle designation.

enter image description here

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2 Answers 2

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You have two equations for $f$ and $g$: $$ f:(4+g)=1:3 $$ (from triangle similarity) and $$ f^2+1=g^2 $$ (from Pythagoras' theorem).

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Another solution is to solve for angle C when you know, for that new dashed-line triangle (the larger one), that hypotenuse ("a") = 5 and one side ("e") = 1. Then use those new angles that are formed from the shift of line "a", and the fact that "e" = 1, to solve for "f" and "g". In any event I end up with the same solution for "f" and "g" when I solve for the equations recommended by Intelligenti Pauca, using the quadratic formula.

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