I have solved the first two problems (1 and 2.1) but 2.2 is giving me trouble. I am able to find some lengths but just come up empty when trying to find the actual answer. If anyone could help me I would be very grateful. Thanks.
2026-03-26 04:51:53.1774500713
Length of an line segment extended from triangle inscribed in rectangle
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Triangles $BAM,BHM$ are congruent, and triangles $MHP,MDP$ are congruent. Also triangles $BAM,MHP$ are similar.
Hence $HP/MH=AM/BA=1/3$, so $HP=2/3$. Hence $DP=2/3$. Also $BP=AH+HP=6\frac{2}{3}$.
Triangles $ABQ,DPQ$ are similar. So $\frac{AB}{BQ}=\frac{DP}{PQ}$ and hence $$\frac{6}{6\frac{2}{3}+PQ}=\frac{\frac{2}{3}}{PQ}$$ and hence $$PQ=\frac{5}{6}$$