Hello! I worked on the following problem for over 30 mins. and still was stuck. Turns out I was totally overthinking it, I knew it was a 30-60-90, but didn't realize the next step was so simple, so I wanted to post to help others.
Original post is here: https://web2.0calc.com/questions/let-t-be-a-point-inside-square-efgh-such-that-te
Let T be a point inside square EFGH such that TE = sqrt6, TF = 2sqrt3, and TG = 3sqrt2. Find angle ETF, in degrees.
Let X, Y, and U be the results of rotating E, H, and T around F by 90 degrees, counter-clockwise, respectively.
Then UF = TF and angle UTF = 90 degrees, so triangle UTF is a 30-60-90 triangle,
UT = TFsqrt2 = 2sqrt6.
ET = EU = TU = sqrt6
1 sqrt3 2
Since it's a 30-60-90,
angle ETF = angle ETU + angle UTF
= 60 degrees + 45 degrees = 110 degrees
As always, thank you to Melody, Alan, CPhil, Asinus, and anyone who I may be missing. This forum is incredibly helpful to my learning!!! I appreciate your help and hard work!!