I am stuck on a problem from a book. I have attached the picture of that question below.
My approach was to draw a line segment between C and X and another line segment between C and Y. This should make triangle XCY. Since XCY is a right triangle (assuming angle XCY is right), I can use pythagorean theorem to find the length of segment XY( answer to the question). Because the lengths of segments CX and CY can be found by using pythagorean theorem on triangle CXD and CBY respectively. The book's solution manual however gives a different answer.
I have explained the author's solution below, in case you require:
The author's approach is to make a segment XL where XL is parallel to GH and the point L lies on CG. Another segment LM which is parallel to FG where the point M lies on segment FB. Then create a triangle XYL. First, the author finds the length of segment YL using pythagorean on right triangle LMY ( point M is to the right of point Y on FB). Length of segment XL is known to be equal to GH. So finally the author, uses pythagorean theorem on triangle XYM to find the answer (ie. the length of segment XY).
While I agree with the author's solution. I don't understand why my solution is wrong and gives a different answer than the author. Is it because my assumption of triangle XCY being a right triangle is wrong? Or something else?
Thanks
Introduce a $3$D reference system with origin in $A$.
Points have coordinates $Y=(4,3,0);\;X=(0,5,3)$
$$XY=\sqrt{(5-3)^2+(3+0)^2+(4+0)^2}=\sqrt{29}$$