Proposed Problem
Click the figure below to see the complete problem 396 about Square, Angle Trisectors, Congruence, Area.
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Complete Problem 396
Level: High School, SAT Prep, College geometry
Sunday, November 29, 2009
Problem 396: Square, Angle Trisectors, Congruence, Area
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1) if extend GF, EF to BC get ang 90°
ReplyDelete=> ang F = 90° ( as vertical angle ) (1)
tr BEF ≡ CFG ≡ DGH ≡ AHE => EFGH rhombus (2)
(1) & (2) => EFGH , square
3) draw PQ//AB through F,J,L,H (P on BC,Q on AD)
name AD = a
find FH diagonal of square EFGH
FH = PQ - (FP+HQ)
HP = (aV3)/2 ( as altitude of equilateral BHC )
=>FP = a - (aV3)/2
=> FH = aV3 - a
=> S EFGH = (d1∙d2)/2
=> S EFGH = ((aV3 - a)(aV3 - a))/2
SEFGH = 2a² - a²V3 (3)
using tr PJC ( and ang 30°) get diagonal of MJKL
=> JL = (3a - a²V3)/3
S MJKL = (2a² - a²V3)/3 (4)
compare (3) & (4) give the result
another way
using S and congruent trapezoids as FPCT (T on CD)
P.S. V3 mean square root of 3