finding the perimeter of a parallelogram inside a right triangle.

Question

 

Triangle ABC is a right triangle.

AC is 20 units long. Points G, H, and D divide AC into 4 congruent parts.

E and F are midpoints.

Find the perimeter of parallelogram DEFG

Answer 1

CG=GH=HD=DA; CG+GH+HD+DA=AC=20, so CG=GH=HD=DA=5.

EF=GD=10=GH+HD.

GĈF=AĈB=EF̂B, so sinAĈB=sinEF̂B=EB/EF=EB/10. Apply Cosine Rule:

In ΔGFC, GF²=GC²+CF²-2GC.CFcosGĈF, GF²=25+CF²-2GC.CFcosGĈF.

cosGĈF=cosEF̂B=FB/10=CF/10 (FB=CF), so 2GC.CFcosGĈF=10CF(CF/10)=CF².

GF²=25+CF²-CF²=25, and GF=5. Since GF=DE, DE=5 and the perimeter of DEFG is 2(GD+DE)=2(10+5)=30.