Let T be a point inside square EFGH such that TE = 1, TF = 2, and TG = 3. Find angle ETF.
+128406 Let T = (x , y)
Let the side of the square = s
We have this system [ we are only interested in s ]
1 = x^2 + ( s - y)^2
4 = x^2 + y^2
9 = ( s - x)^2 + y^2
This is a little sticky to solve, so using WolframAlpha, we get that s = √ [ 5 + √8]
Using the Law of Cosines we have that
s^2 = ET^2 + FT^2 - 2(ET * FT) cos ETF
5 +√8 = 1^2 + 2^2 - 2(1*2) cos ETF
5 + √8 = 5 - 4 cos ETF
√8 / [ -4] = cos ETF
cos ETF = - 2√ 2 / [ 2 * 2] = -√2/2 = -1/√2
arccos (-1/√2) = ETF = 135°
