I actually wanted to try method that doesn't involve Quadratic Equation, but because I was in a hurry and already I have seeing the method I used the moment I looked at the Question, I just thought it was also a perfect solution as well. But this method of yours is the best, thanks
Thank you very much for the problem and its solution. I think, an alternative solution would be as follows: As the angles AEB and CED are equal, one can exploit this by means of similarity. For doing so, one can extend EB in the direction from E to B to obtain the point "P" such that the angle APE is equal to 90°. Triangles APE and CDE are then similar and the points E, D, P and A all lie on the circle with center at the midpoint of AE and with radius |AE|/2. If one calls |EB| = m, one then has that |EB|*|BP| = |AB|*|BD| = 36 = m*|BP|, i.e. |BP| = 36/m. Furthermore, in triangle EBD one has by Pythagoras that |DE| = sqrt(m^2-36). By similarity of the triangles BAP and BED one has that |AP| = sqrt(m^2-36)*6/m and by similarity of the triangles EPA and EDC one has that sqrt(m^2-36)/2 = (m+36/m)/(6*sqrt(m^2-36)/m), which is equivalent to 3*m^2 - 108 = m^2 + 36, i.e. m^2 = 72 and sqrt(m^2-36) = 6 = |DE|. From this, it follows that [BCE] = |BC|*|DE|/2 = 4*6/2 = 12.
@@MathandEngineering same as above: too much, too many details. if smn complains they dont get it, they should go to lower grade and practice quick thinking.
you are using all small letters for everything, lentghs, areas, angles. this is so confusing. why not use greek letters for angles, and large letters for areas? or capital S with subscript for the respective triangle surface area. then we see immediately that y=3z, and y^2 = 9z^2. you can make it shorter. thanks
Wow I looked for this Method, but I didn't give it enough time, I also prefer methods that don't involve any Quadratic or Cubic Equation, that you so much for this amazing solution
Actually my goal was to make it simpler, but at least if someone doesn't know that 288 is the same as -(-288), then they'll know from the video, Why I do this is because I don't want to skip anything, because someone will also complain, They'll be like, how did you arrive at this?, I believe its best to include everything than assuming, the audience know and skip it
@@MathandEngineering again, you are explaining high-level ideas, like for very low level folks. ppl who cant figure out things faster, they will never grasp the idea of the whole prove. I am bored by the overly detailed derivation, and loose my focus.
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I solved the problem by the tangent rule. [BCE] = BC*DE/2 = 4r/2 = 2r. To find r, let the middle angle in the figure be "b". Then tan (2a+b) = 12/r = [tan (a+b) + tan (a)] / [1-tan (a+b)*tan (a)] = [6/r+2/r] / [1-6/r*2/r] = [8/r] / [1-12/r^2] = 8r/ [r^2 - 12] = 12/r. Then 12(r^2-12) = 8r^2 --> 12r^2 - 144 = 8r^2 --> 4r^2 = 144 -> r^2 = 36 -> r=6m. Then [BCE] = 2*6 = 12 m^2
I actually wanted to try method that doesn't involve Quadratic Equation, but because I was in a hurry and already I have seeing the method I used the moment I looked at the Question, I just thought it was also a perfect solution as well.
But this method of yours is the best, thanks
Congratulations @juanalfaro7522. As you can see, i made a quite similar method before you.
Thank you very much for the problem and its solution. I think, an alternative solution would be as follows: As the angles AEB and CED are equal, one can exploit this by means of similarity. For doing so, one can extend EB in the direction from E to B to obtain the point "P" such that the angle APE is equal to 90°. Triangles APE and CDE are then similar and the points E, D, P and A all lie on the circle with center at the midpoint of AE and with radius |AE|/2. If one calls |EB| = m, one then has that |EB|*|BP| = |AB|*|BD| = 36 = m*|BP|, i.e. |BP| = 36/m. Furthermore, in triangle EBD one has by Pythagoras that |DE| = sqrt(m^2-36). By similarity of the triangles BAP and BED one has that |AP| = sqrt(m^2-36)*6/m and by similarity of the triangles EPA and EDC one has that sqrt(m^2-36)/2 = (m+36/m)/(6*sqrt(m^2-36)/m), which is equivalent to 3*m^2 - 108 = m^2 + 36, i.e. m^2 = 72 and sqrt(m^2-36) = 6 = |DE|. From this, it follows that [BCE] = |BC|*|DE|/2 = 4*6/2 = 12.
10.23: please please simplify! divide the whole thing by -8=/=0 and use smaller numbers!!!
10:46 is a monstrous expression. could have been much simpler
Please what is Wrong with the Equation? I just checked it and I am confused as to why you said it's monstrous
@@MathandEngineering same as above: too much, too many details. if smn complains they dont get it, they should go to lower grade and practice quick thinking.
10:09 there should be no r in your equation. -8x+144x+5184=0. no r
Oh no, sorry please i didn't notice that, it's a typo, thanks you so much
@@MathandEngineering thats why we are here, to proof-read.
you are using all small letters for everything, lentghs, areas, angles. this is so confusing. why not use greek letters for angles, and large letters for areas? or capital S with subscript for the respective triangle surface area. then we see immediately that y=3z, and y^2 = 9z^2. you can make it shorter. thanks
Thank you so much for this amazing advice, I promise I'll look into it and improve, "Math and Engineering" Needs people like you thanks 🙏
area BCE = BC*DE/2=4*r/2=2*r
Let's call angle (BEC) = b
tan(a)=2/r
tan(a+b)=(4+2)/r=6/r
tan(2*a+b)=(6+4+2)/r=12/r
tan(2*a+b)=tan(a+(a+b))=(tan(a)+tan(a+b))/(1-tan(a)*tan(a+b))
12/r=(2/r+6/r)/(1-2/r*6/r)
12/r=(8/r)/(1-12/r^2)
12=8/(1-12/r^2)
1-12/r^2=8/12=2/3
12/r^2=1-2/3
12/r^2=1/3
r^2=3*12=36
r=6
area BCE = 2*r = 2*6 = 12 m²
Wow I looked for this Method, but I didn't give it enough time, I also prefer methods that don't involve any Quadratic or Cubic Equation, that you so much for this amazing solution
@@MathandEngineering the video also goes tru quadratic, except it shows the longest way, in exclusive details.
i love your way. simple, short elegant.
ctg(α+β)=(ctgα•ctgβ-1)/(ctgα+ctgβ)
11:40 why so complicated: everyone knows that 288/-16 is negative and has no meaning. you are overcomplicating your explanation.
Actually my goal was to make it simpler, but at least if someone doesn't know that 288 is the same as -(-288), then they'll know from the video,
Why I do this is because I don't want to skip anything, because someone will also complain,
They'll be like, how did you arrive at this?, I believe its best to include everything than assuming, the audience know and skip it
Though, I disagree with the part you said "Everyone knows that 288/-16 is negative"
@@MathandEngineering again, you are explaining high-level ideas, like for very low level folks. ppl who cant figure out things faster, they will never grasp the idea of the whole prove. I am bored by the overly detailed derivation, and loose my focus.