\[tan 45=\frac{PB}{PG} \rightarrow 1=\frac{PB}{4}\rightarrow PB=4\]
Then
\[AB=x+4\]
By syymetry,
\[EF=x+4\]
and then \[FG=10-(x+x+4)=6-2x\]
\[\frac{1}{2}(x+x+4)(4)=(6-2x)(4) \rightarrow x+2=6-2x \rightarrow 3x=3x=4 \rightarrow x=\frac{4}{3}\]
How to Cut a Rectangle in Three With Parallel Lines to Give Three Equal Areas
Suppose we have a rectangle cut into three by two parallel lines BG and CF making an angle of 45° with EG as shown. The areas of the three shapes formed are equal, so that the area of ABGH, BCFG and CDEF are equal. What must the length of HG be so that this is so?
Then
By syymetry,
Then
By syymetry,