Find the blue shaded area | Geometry Problem | 2 Different Methods | Very Important Geometry Skills

[@brandonk9299]

If you rotate this triangle, you can apply the ladder theorem which relates the areas: 1/(AEF+AFB) + 1/(ADB+AFB) = 1/ABC + 1/AFB. So 1/(33+44) + 1/(22+44) = 1/T + 1/44. T is the total area of the triangle, and we find T=184.8. The area of CDFE= T-(22+33+44)= 184.8-99=85.8

[@davidellis1929]

If you know Mass Points, you can solve as follows. Let G be the point where AD and BE meet. Since AG:GF=44:22=2:1, assign weights A=n, D=2n. Similarly assign B=3m, E=4m. The weight of C is 2n-3m along BDC and 4m-n along AEC. Then 2n-3m=4m-n, so 3n=7m. If we set n=7, m=3 then the weights of A and C are 7 and 5. Thus the area of BEC is (7/5)*77, or 107.8. Now subtract 22 to yield the blue area of 85.8.

[@SumitVerma-lg3qh]

It can be very easily solved by using area lemma and menalaus theorem Let required area is x In triangle ABE =>AF :FE= 4:3. In traingle ABD and DAC=> BD:DC=66:33+x In triangle ABE and BEC => CE:EA=22+x:77 Using menalaus theorem in triangle BEC , 66/33+x * 99+x/77 * 3/4=1 Which gives x as 85.8 💁👀

[@gabrielfreitas608]

Hello, I'm from Brazil and I'm really enjoying your channel. I would like you to prove the following: (a - 1)^a < a^(a - 1), where "a" is a positive real number greater than 1,

[@hunghinsun2123]

Your statement can be disproved by considering a counter example when a=4.

[@gabrielfreitas608]

@@hunghinsun2123 It's the opposite? (a - 1)^a > a^(a - 1

[@hunghinsun2123]

By using the Newton-Raphson Method, the root of (x-1)^x-x^(x-1)=0 can be found to be 3.293166 (denoted by b). It is claimed that (x-1)^x > x^(x-1) for x>b The inequality can be rewritten as ln(x-1)/(x-1) > lnx/x. Consider the function f(x)=lnx/x and f'(x)=(1-lnx)/x^2. Then f(x) is increasing for 0=1+e as f(x) is decreasing in this range. For the range b

[@sauravmalik9025]

​@@gabrielfreitas608no

[@Alessandro-1977]

Interesting, so exciting ! 👏👏👏

[@oleg91527]

Great explanation!!!

[@skwest]

Brilliant.

[@kokawiliam1258]

Very nice

[@8mice979]

you could just use Menelaus in the ACD triangle

[@thanakornkittikomolpong5088]

Why .... i cal.. it to be 55