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

Labels:
area,
congruence,
square,
trisection

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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