Find the Area of the Red Triangle | A Rectangle and 3 Triangles #geometry #maths

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[timc]

tan(DEA) = 6/2 = 3
tan (DEB) = tan( DEA + 45) = [3 + 1]/[1 - (3)(1)] = -- 2,
tan(BEC) = tan(180 - DEB) = -- tan(DEB) = 2 = 6/(EC),
so EC = 3, and area = (1/2)(6)(3) = 9

[quigonkenny]

Let α = ∠DEA, β = ∠BEC, EC = h.

tan(α) = 6/2 = 3
α = arctan(3) ≈ 71.565°

β = 180 - 45 - α ≈ 63.435°

tan(63.435) = 2

6/h = 2
h = 6/2 = 3

A = bh/2 = 6(3)/2 = 9

[plamenpenchev]

Draw a horizontal line from E right dividing 45° angle. Angles are a + b = 45, a = EAD, b = EBC.
Then tan(a) = 2/6, area = 6×6×tan(b)/2.
tan(b) = tan(45 - a) =
(tan(45) - tan(a))/(1 + tan(45)×tan(a)) =
(1 - 2/6)/(1 + 1×2/6) = 1/2,
then area = 9

[phungpham]

1/ EA=2sqrt10 (it's easy). Just drop the height BH=h to EA, because the angle DAE=ABH ( having sides perpendicular) so the 2 triangles EDA and AHB are similar----> ED/DA=AH/BH=2/6=1/3
We have EH = h and HA= h/3 h/3=HA/1=(EH+HA)(3+1)= EA/4----> h=3EA/4= 3/4 . 2sqrt 10 = 3/2 sqrt10
2/The area of the AEB = half of the area of the rectangle
so the area of the red triangle= Area of the AEB- Area of the triangle EDA
= 1/2x 3/2 sqrt10 x 2 sqrt10 - (1/2x 2x6) = 15-6=9 sq units

[TristanPopken]

a=arctan(2/6)
s=tan(45-a)
notice that s=0.5, so 6*3/2=9

[hussamaldeenzakaria]

You can easily solve
If drew line from E parallel DA intersect AB at k
Then angle Tan DAE =1/3 that equal to tan KEA…then Tan KEB =tan (45-AEK)=(1-1/3)/(1+1/3)=1/2, then KB segment =1/2*6=3—-> then area of red triangle =1/2*3*6=9

[michaelmccabe-wu]

It’s much easier. Equilateral triangle with angle 45 deg means that there are 4 such triangles in a rectangle. Area of the rectangle is 12. Divide by 4 and y have your answer

[KipIngram]

Obviously, without doing a single "real" calculation, it's 3. Total area is 12 and that shaded part is a quarter of it. Easy peasy.

[dayletba]

it is even possible to do it with sharp eyes: if the height is 6, and base is 2, by looking at the base of the second triangle ( what we are looking for) you can definitely see that it is less that 6 and more than 2, so 3

the formula 1/2(b)(h) so 1/2(6)(3)= 18/2= 9

thats it

but i really appreciate your work bro, i am not able to solve this like that, useful👏

[jesusantonioviancharangel]

Muy complicado, pudiendo hacerlo más sencillo.

[mrmimi]

Just use pathogryan theoreme to calculate AE then use the cos and sin to calculate the angles then use the sin in the other triangle to get the result

[johnspathonis]

I got the same answer by using the rule :- Cos(A-B) = Cos(A).Cos(B) + Sin(A).Sin(B) Firstlty drop a perpendicular from E to AB to a point F. Angle A is 45 and angle (B) is the angle DAE which is equal to the angle AEF. Once the Cos (A-B) is found the distance EB is found. Use pythag to find EC etc. Much less mathematics.

[dakcom-mkmp]

You just need to find the angles by tangent of E, and then find x to calculate the area

[karamsedighi]

Check please if you draw b to d ??? tang 45= 5/6 no no no