#Is there a faster way to do this (than cosine rule, then sine rule, then cosine rule)?
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the angle bisector theorem can be used if we know that the 45 degrees is cut in half into 2 exact angles of 22.5, and I do not know if that is the case
but then u get 2x/2 = 3x/5 which makes x=0, so I don't think angle CAP = PAB
no, 2/5 ≠ 2x/3x so its not an angle bisector