Geometry Problem

Click the figure below to see the complete problem 516 about Triangle, Cevian, Congruence, Angles.

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Complete Problem 516

Level: High School, SAT Prep, College geometry

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## Monday, August 30, 2010

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Problem 516: Triangle, Cevian, Congruence, Angles

See also:

Complete Problem 516

Level: High School, SAT Prep, College geometry

Online Geometry theorems, problems, solutions, and related topics.

Geometry Problem

Click the figure below to see the complete problem 516 about Triangle, Cevian, Congruence, Angles.

See also:

Complete Problem 516

Level: High School, SAT Prep, College geometry

Labels:
angle,
cevian,
congruence,
triangle

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x=11.25deg. (4x=45deg).

ReplyDeleteLet me give some supplement:

ReplyDeleteLet point E, AE//DB and AD//EB.

Triangle ACE is isosceles, angle EAC = angle BDC.

So, angle AEC = angle ACE = 5x, that is, angle ECB = x.

Angle EAB = x, since AE//DB.

EA=BC, since points A, C, B, E are on a circle, and AC//EB.

So, triangle ABE and CEB is congruence, that is,

CB = EA = BD, so triangle DCB is isosceles ---> angle BDC is 6x.

6x + 6x + 4x = 16x = 180deg. ---> x = 11.25deg.

Draw AE//BD with AE=BD

ReplyDeleteADBE is a parallelogram and AEC isosceles . angle (EAC)=180-10x , angle(AEC)=5x

AEBC is a cyclic quadrilateral with BE//AC so AEBC will become isosceles trapezoid with AE=BC

Angle(BAC)=angle (ECA)=5x

In triangle ABC we have 5x+5x+6x=180 so x=11.25

Peter Tran

Mark E on AC extended such that < CBE = x. Then DB = DE = AC so AD = CE

ReplyDeleteNow draw circle BDE to cut AB at P and BE at Q. Since equal arcs subtend equal angles in a circle PD = CQ which in turn subtend equal angles at Q and D showing that PQ//DC

So PDCQ is an isoceles trapezoid and Tr.s APD & CQE are congruent making Tr. ABE and hence Tr. BDC both isoceles. So < BDC = 6x and therefore 6x + 6x + 4x = 180 and x = 11.25

Sumith Peiris

Moratuwa

Sri Lanka

Problem 516

ReplyDeleteLet E symmetry of D on the AB, then EB=BD=AC and <EBA=<DBA=x and <AEB=<ADB=10x.But

<EBC=6x=<ACB then the AEBC is isosceles trapezoid (ΑΕ//ΒC).So <AEB+<EBC=180 or 10x+6x=180 therefore x=11.25

APOSTOLIS MANOLOUDIS 4 HIGH SHCOOL OF KORYDALLOS PIRAEUS GREECE