## Sunday, May 18, 2008

### Elearn Geometry Problem 3

See complete Problem 3
Triangle, Angles, Median, Congruence. Level: High School, SAT Prep, College geometry

1. Solution to this posted at http://www.osinfofrom.us/prob003.html

Why "ABC appears to be a right triangle"?
Why triangles ABD and DBC are congruent?

3. For lack of better ideas, here's a solution by analytical geometry: Let D be the origin and A:(-1,0) & B;(1,0) and let angles BAC & BCA be 3m & m resply. Also let tan(m)=t. Now BD:y=-x while BC:y=(1-x)t. Thus, B is (t/(t-1),-t/(t-1)). Since co-ordinates of A & B are known we've Slope BA = t/(1-2t) = tan(3m) = (3t-t^3)/(1-3t^2)
In other words, (1-2t)(3-t^2)= (1-3t^2) which yields three solutions of which the only admissible solutions is t=tan(m)=(2)^(1/2) - 1
In other words, m=angle BCA = 22.5 deg.
Ajit:ajitathle@gmail.com

4. Attempted second explanation at http://www.osinfofrom.us/prob003.html as Antonio pointed out a flaw in the first attempt to prove the value of x.

5. Are Joe and Ajit synonymous?

6. They're the same person, not synonymous though!
Ajit

7. If you want to use trigonometry then:
In Tr. ABC -- AC/sin(3x+x)= AB/sin(x)
or AC/AB =sin(4x)/sin(x)
In Tr. ABD -- AD/sin(3x+45)= AB/sin(45)
or AC/AB=2sin(3x+45)/sin(45)
or sin(4x)/sin(x)=2sin(3x+45)/sin(45)
= 22.5 degrees as the only admissible solution.

8. http://geometri-problemleri.blogspot.com/2009/05/problem-12-ve-cozumu.html

9. Chose point E on BC so that CAE = x. BEA is exterior angle of AEC so that BEA = 2x and AB = BE. Let F is midpoint of AE clearly triangles AFB and EFB are congruent. If H and G are midpoint of AB and BE respectly then AHF, BHF, BGF and EGF are congruent so that HAF = HBF = GBF = GEF = 2x.
So finally x + x + 2x + 2x + 2x = 8x = 180 deg  x = 22.5 deg.
wisnab36@gmail.com

10. Why are AHF, BHF, BGF and EGF congruent?

11. Thank’s Mr. Antonio for your comment. I will try to justify that.
Triangle AFB is right at F and HF is median sothat HF = AH = HB. By the same reason we can find that GF = BG = GE.
AFB and EGF are equilateral triangles so that anggle AFH = anggle EFG = 2x.
From these facts, HF // BE and AB // BG so that BGFH is a rhombus which has properties HG = BF ……………………………. (1)
In ABE triangle, because H and G mid points of AB and BE respectly then HG // AE and HG = ½ AE = AF ………………… (2)
From (1) and (2) we get BF = AF = FE ……………………….. (3)
From (3) anggle ABF = anggle EBF = 2x.
We had showed that HB = HF = BG = GF
From these we get angle HFB = angle BFG so that AHF, BHF, BGF and EGF congruent

From (3) we can find another solution.
Because F is midpoint of AE and AF = FE = BF then ABE = 90 deg.
So that 2x + x + x = 4x = 90 deg  x = 22.5 deg
Thank’s
wisnab36@gmail.com

12. angle ABD=135-3x & DBC=45-x
so angle ABD=3z & DBC=z
area of ABD= area of DBC there fore
1/2AB.BD.sin3z=1/2BC.BD.sinz
so sin3z/sinz=BC/AB
due to sine theorem in ABC sin3x/sinx=BC/AB
there fore sin3x/sinx=sin3z/sinz
therefore 3-4sin^2(x)=3-4sin^2(z)
x and z are oblique so x=z
in triangle ABD 3x+3x+45=135 so x=22.5
please tell me how to send another picture solutions?(henrik_ir@yahoo.com)

13. Correção: de Valdir Marques Faria

A resolução de Anonymou não é consistente quando justifica a congruência dos triângulos. O ângulo de 45° é determinante na resolução do problema. Assim, teremos:
No triângulo BAD, triângulo ABD = 135°– x e, no triângulo BDC, triângulo DBC = 45°+x. Aplicando a lei dos senos nos triângulos ABD e BDC, teremos:
sen(3x)/sen(135°-3x) = sen(x)/sen(45°-x) ...
sen(3x).sen(45°-x) = sen(135°-3x).senx ...
sen(3x).(sen45°.cosx – cos45°.senx) = senx(sen135°.cos(3x) – sen(3x).cos135°) ...
sen(3x).(cosx – senx) = senx.(cos3x + sen3x) ...
sen(3x).cosx – cos(3x).senx = 2.sen(3x).senx ...
sen(2x) = cos(2x) – cos(4x) ...
sen(2x) – cos(2x) = (sen(2x) + cos(2x)).(sen(2x) – cos(2x))
Se sen(2x) for diferente de cos(2x) teremos:
sen(2x) + cos(2x) = 1 , e assim x = 0 ou x = 45° o que não convém.
Se sen(2x) = cos(2x), teremos 2x = 45 e x = 22,5°.

14. I'm brazilian , and my english i'snt good,but i saw a beautifull solution for it, really fast.However i'll explain in portuguese, please translate it.
Pronlogando o lado AB, traçamos uma reta partindo de C, quer corte perpendicularmente o prolongamento do lado AB, fprmando o ponto E.Se traçarmos o segmento ED, este será a emediana relativa a hipotenusa do triângulo retângulo ACE, portanto terá a mesma medida de AD.Tendo o ângulo C como 90-3x, aparecerá o triângulo isósceles CDE cujo ângulo EDC também será 90-3x.
Portanto, 90-3x+90-3x+135=180, logo = -6x=-135, portanto x=22,5

15. The solution of mr. Anonymous (wisnab36@gmail.com) is nog correct!
He says:
"From these facts, HF // BE and AB // BG so that BGFH is a rhombus which has properties HG = BF ."
That's NOT true!
Yes, it's a rhombus, but NOT HG = BF!!!
It's also cleat that his proof can't be correct, because he doesn't use the fact that angle ADB = 45°. IF his proof was solid, then it would be true voor EVERY triangle with angles x and 3x, that x = 22,5°.

16. Build quadrilateral ABEC recordable => [DB - bisector [EB - bisector x = 22.5

Greetings

18. Video solution (Spanish version) by Eder Contreras and Cristian Baeza at Geometry problem 3. Thanks Eder.

19. http://henrik-geometry.webs.com/3.htm

20. Select Point E at BC so that EAC=x
So, BEA=2x
therefore AB=BE (as BEA=BAE=2x)
Since EAC=ECA=x
While Drawing a perpendicular to AC from E it meets AC at D
Now Draw a perpendicular From B to AC,
Say it is BG
Since Tri.BGD is 90-45-45
BG=GD
Now Draw a parallel to DG through E, so it meets BG at F
so FE=GD
ie:BG=EF
BEF=ECD=x (Corresponding angles, EF//AC)
so Tri BAG and Tri.BEF are congruent
therefore ABG=x
GBD=45, DBC=45-x
So, ABC=x+45+45-x
ABC=90 Deg.
Now draw a semicircle having its center D and radius DA.
it passes through A,B,C as AD=DC and ABC=90Deg
SInce the angle subtended at center =2.angle subtended at circumference
BDA=2.BCA
2.BCA=45
2x=45Deg,
so, x=22.5 Deg.

Very nice Question, Thanks Mr.Antonio Gutierrez.
.................

21. Problem 003 revisited.
Draw through B a parallel to AC and trough D a perpendicular do AC. These two lines meet at E, and DE meets BC at G.
If BF is the altitude of ABC, then BEDF is a square.
We have AG = CG, then ang(GAC) = x and ang(BGA) = ang(BAG) = 2x.
So ABG is isosceles with AB = BG and triangles ABF and GBE are congruent.
Since ang(BAF) = 3x and
ang(BAF) = ang(BGE) = ang (CGD) = 90 – x,
then 3x = 90 – x, or x = 22,5.

22. Drop BN ⊥ AC
cot x - cot 3x = (NC - NA)/BN = 2ND/BN = 2
sin 3x cos x - cos 3x sin x = 2 sin 3x sin x
sin 2x = 2 sin 3x sin x = cos 2x - cos 4x
cos 4x = cos 2x - sin 2x
cos²2x - sin²2x = cos 2x - sin 2x
(cos 2x - sin 2x) (cos 2x + sin 2x) = (cos 2x - sin 2x)
cos 2x + sin 2x = 1 or cos 2x - sin 2x = 0

If sin 2x = 1 - cos 2x then
2 sin x cos x = 2 sin²x
cos x = sin x, x = 45° not admissible

Hence cos 2x - sin 2x = 0
tan 2x = 1,
2x = 45°,
x = 22.5°

23. Mi solución con geo clasica :p (en español) http://i50.tinypic.com/2jfalbp.png

24. Let ED be the perp bisector of AC, E being on BC. Perp.s from A and E to BD are equal hence Tr ABD is congruent to Tr EBD and the result follows.

Sumith Peiris
Moratuwa
Sri Lanka

25. I have a bad English. See Whether you could understand my solution.
Mark a point E on the side BC such that angle BAE = angle BEA = 2x
so that BE = BA
Since Tri AEC is isosceles , the perpendicular drawn from E to BC meet the side BC at D.
This implies angle BDE = 45 deg.
Now we shall extend the line DE to F such that DF = AD
it's obvious that Tri BAD congruent to Tri BDF (S.A.S)
This implies BF = BA = BE and angle BFD = 3x
In Tri BEF, angle BEF = angle BFE = 3x
angle BEF = angle DEC = 3x (vertically opposite angles)
So, in Tri DEC 3x + x = 90 degree
4x = 90 degree
therefore x = 22.5 degree

27. @NagaratnamKrishnakanth
Nice solution.

28. Construct point E, where AE=AC and E lies on BC produced. Hence, triangle ACE is an isosceles triangle where <AEC=<ACE, hence <AEC=x. Let AE//DB.

Since AE//DB and <DAE is 135° (<DAE+<EAD=180°), <BAF=135°-3x
On the other hand, by angle sum of triangle AEC, <BAF can also be expressed as 180°-5x.
Hence, 180°-5x = 135°-3x
45°=2x
x=22.5°
Proved

29. 2nd Solution

Let E be BC such that < EAC = x so that < AEB = 2x = < BAE
Let F be mid point of AE so that < ABF = < EBF = 90-2x
But < CBD = 45-x = < DBF = < FDB since FD//BC and hence < ADF = x
Therefore AF = FD = FE = FB
Hence AFB is a right isoceles triangle and so 2x = 45 and x = 22.5

Sumith Peiris
Moratuwa
Sri Lanka