Here’s my version with trigonometry to share with. tan a=12/x tan 2a=(22+15)/x=27/x tan 3a=(12+15+y)/x=(27+y)/x 27/x=tan 2a=(2*tan a)/(1-tan(a)^2)=(2*12/x)/(1-(12/x)^2)=24x/(x^2-144) cross multiplying 27*x^2-27*144=24*x^2 -> 3*x^2=144*27 ->x^2=144*9 -> x=36 tan a=12/x=12/36=1/3 tan 2a=27/x=27/36=3/4 (27+y)/x=(27+y)/36=tan 3a=(tan a+tan 2a)(1-tan a*tan 2a)=(1/3+3/4)/(1-1/3*3*4)=13/9 Cross multiplying 27+y=52 ->y=25 Area ADE=1/2*y* x=1/2*36*25=450
Well done! I planned to do it this way, but you beat me to it. Some steps are not shown, so I'll fill them in to make it easier to follow. x and y are defined in the video. Basically, the equation beginning with 27/x is the tangent double angle formula, tan(2Θ) = (2 tan(Θ)/(1 - tan²(Θ)) with tan(2Θ) replaced by 27/x and tan(Θ) by 12/x, and it is solved for x = 36. The equation beginning with (27+y) is the tangent sum of angles formula tan(α + ß) = (tan(α)(tan(ß)/(1 - tan(α)tan(ß)) with α replaced by a and ß by 2a, so α + ß becomes 3a. tan(3a) = tan(a + 2a) = (27 + y)/x = (27 + y)/36 = 3/4 + y/36, tan(a) = 12/36 = 1/3, tan(2a) = 27/x = 27/36 = 3/4. Solving, y = 25. y is the base of ΔADE and x is the height, so area = (1/2)yx = (1/2)(25)(36) = 450.
This problem was posted some time ago. I used the same method and obtained the same answer as @xualain3129 of DE = 25, AB=36, and [ADE] = 36*25/2 = 450 m^2
Wow you have a good memory, I thought no one would recognize it, but you caught me😂😂😂😂, I actually wanted to solve it with a clearer method, because in the other video, it was a trigonometric approach I used, so I knew most of the times when people see its the same Question, they just skip becuase they think they are repeating the same video, and they'll miss the update without knowing, so i made some changes so that it be a new Question and a new solution. But don't tell anybody 😄😄
Congrats! It is always helpsull to get several ways to deal with slicing tringles.
Yes sir, thank you so much
Here’s my version with trigonometry to share with.
tan a=12/x
tan 2a=(22+15)/x=27/x
tan 3a=(12+15+y)/x=(27+y)/x
27/x=tan 2a=(2*tan a)/(1-tan(a)^2)=(2*12/x)/(1-(12/x)^2)=24x/(x^2-144) cross multiplying
27*x^2-27*144=24*x^2 -> 3*x^2=144*27 ->x^2=144*9 -> x=36
tan a=12/x=12/36=1/3
tan 2a=27/x=27/36=3/4
(27+y)/x=(27+y)/36=tan 3a=(tan a+tan 2a)(1-tan a*tan 2a)=(1/3+3/4)/(1-1/3*3*4)=13/9
Cross multiplying
27+y=52 ->y=25
Area ADE=1/2*y* x=1/2*36*25=450
Wow it's so simple with trigonometry, thank you so much for sharing
Well done! I planned to do it this way, but you beat me to it. Some steps are not shown, so I'll fill them in to make it easier to follow. x and y are defined in the video. Basically, the equation beginning with 27/x is the tangent double angle formula, tan(2Θ) = (2 tan(Θ)/(1 - tan²(Θ)) with tan(2Θ) replaced by 27/x and tan(Θ) by 12/x, and it is solved for x = 36. The equation beginning with (27+y) is the tangent sum of angles formula tan(α + ß) = (tan(α)(tan(ß)/(1 - tan(α)tan(ß)) with α replaced by a and ß by 2a, so α + ß becomes 3a. tan(3a) = tan(a + 2a) = (27 + y)/x = (27 + y)/36 = 3/4 + y/36, tan(a) = 12/36 = 1/3, tan(2a) = 27/x = 27/36 = 3/4. Solving, y = 25. y is the base of ΔADE and x is the height, so area = (1/2)yx = (1/2)(25)(36) = 450.
I used a trig approach as well. No calculator but I did have to look up the multiple angle formulas!
Essa foi uma das questões mais lindas que já vi !
Ohh sir, I am really delighted be this comment of yours, thank you so much
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This problem was posted some time ago. I used the same method and obtained the same answer as @xualain3129 of DE = 25, AB=36, and [ADE] = 36*25/2 = 450 m^2
Wow you have a good memory, I thought no one would recognize it, but you caught me😂😂😂😂, I actually wanted to solve it with a clearer method, because in the other video, it was a trigonometric approach I used, so I knew most of the times when people see its the same Question, they just skip becuase they think they are repeating the same video, and they'll miss the update without knowing, so i made some changes so that it be a new Question and a new solution.
But don't tell anybody 😄😄