The transformation is ISOMETRY.
In mathematics, an isometry<span> (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. A composition of two opposite </span>isometries<span> is a direct </span>isometry<span>. A reflection in a line is an opposite </span>isometry<span>, like R </span>1<span> or R </span>2<span> on the image.</span>
complementary angles are angles that, when added together, make 90 degrees.
Use the identity: 
and solve for 
. You get: 
Do the substitution on the left side to get:
There is a not so well-known theorem that solves this problem.
The theorem is stated as follows:
"Each angle bisector of a triangle divides the opposite side into segments proportional in length to the adjacent sides" (Coxeter & Greitzer)
This means that for a triangle ABC, where angle A has a bisector AD such that D is on the side BC, then
BD/DC=AB/AC
Here either
BD/DC=6/5=AB/AC, where AB=6.9,
then we solve for AC=AB*5/6=5.75,
or
BD/DC=6/5=AB/AC, where AC=6.9,
then we solve for AB=AC*6/5=8.28
Hence, the longest and shortest possible lengths of the third side are
8.28 and 5.75 units respectively.
