Proposed Problem

Click the figure below to see the complete problem 424 about Triangle, Cevian, Angle, 90 degree, Congruence.

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Complete Problem 424

Level: High School, SAT Prep, College geometry

## Tuesday, February 9, 2010

### Problem 424: Triangle, Cevian, Angle, 90 degree, Congruence

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http://geometri-problemleri.blogspot.com/2010/02/problem-70-ve-cozumu.html

ReplyDeleteLet be m(ACB)=y and m(BAC)=2y

ReplyDeletein triangle ABC: 5y+x=90

knowing y is knowing x,and 0<5y<90

with the sine rule in ABD and ABC

BD=AB.sin2y/cos3y;BC=AB.sin2y/sin3y

in the right triangle BCD

BC²+BD²=DC²=AB²

then sin²(2y)=cos²(3y).sin²(3y)=sin²(6y)/4

we have the formula: sin(3a)=3sina-a(sina)^3

with t=2y ,sin²t=1/4

2y=30

x=15

.-.

Let E be the mid point of CD and F the mid point of AB. Then E is the centre of right Tr. BCD and so BF = BE.

ReplyDeleteNow < ABE = 180 - 2/3<C - 2<C = 180 - 8/3< C. Hence < BFE = < BEF = 4/3<C.

Hence < FEA = 4/3<C - 2/3<C = 2/3<C = < FAE. So Tr. BEF is equilateral and BE is perpendicular to AC and so < C = 45 and it follows that x = 15

Sumith Peiris

Moratuwa

Sri Lanka

Let x+2y+3y=90

ReplyDeletex=90-5y

<BDA=180-(90-5y)-2y=90+3y

In triangle BAD

sin(90+3y)/AB=sin2y/BD

BD/AB=sin2y/cos3y-------(1)

In triangle BCD

sin3y=BD/DC-------(2)

Since AB=CD, (1)=(2)

sin2y/cos3y=sin3y

sin2y=sin3ycos3y

2sin2y=sin6y

sin6y/sin2y=2

sin3u/sinu=2, with u=2y

3-4(sinu)^2=2

sinu=1/2

u=30, y=15

x=90-5y=15