Geometry Problem

Click the figure below to see the complete problem 484 about Square, Angle, 90 degrees, Triangle, Measurement, Proportion.

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Complete Problem 484

Level: High School, SAT Prep, College geometry

## Wednesday, July 28, 2010

### Problem 484: Square, Angle, 90 degrees, Triangle, Measurement, Proportion

Labels:
90,
degree,
measurement,
proportions,
square,
triangle

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Denote (XYZ) =angle XYZ

ReplyDeleteWe have ( ABC)=(AEC)=(ADC)=90

So polygon ABECD is cyclic with AC as a diameter of circumcircle of the polygon

In this circumcircle , arc BA=arc AD=arc DC= ¼ of full circle = 90

(BEA)=(AFD)=(DEC)= 45 ( These angles face 90 degrees arcs )

Consider triangle BEG and angle BEG, EF is internal angle bisector of ( BEG)

So FB/FG=EB/EG ( property of internal angle bisector)

Since EC perpen. To EF , EC will be external angle bisector of (BEG)

And CB/CG= EB/EG ( property of external angle bisector)

So FB/FG= CB/CG

Peter Tran

Problem 484

ReplyDeleteIs ABECD cyclic (<AEC=90=<ABC).Then <BEA=45=<AED=<DEC.So EG is internal bisector and the EB is external bisector of triangle FEC.Then FG/GC=EF/EC and EF/EC=BF/BC.

So FG/GC=BF/BC or BF/FG=BC/CG.

APOSTOLIS MANOLOUDIS 4 HIGH SHCOOL OF KORYDALLOS PIRAEUS GREECE

Triangles BEG and CDG are similar so

ReplyDeleteBE/EG = CD/CG.....(1)

EF bisects < BEG hence

BE/EG = BF/FG...(2)

From (1) and (2) BC/CG = BF/FG

Sumith Peiris

Moratuwa

Sri Lanka

From similarity of ABF and CEF we get

ReplyDeleteBF/AB=EF/EC ........(1.)

From similarity of EFG and AED we get

EF/FG=AE/AD ........(2.)

From similarity of AEC and DCG we get

GC/CD=EC/AE ........(3.)

Multiplying equation 1 and 2 with AB=AD gives :

BF/FG=AE/EC

And from equation 3, with CD =BC we get:

BF/FG=CD/GC

BF/FG=BC/GC