Triangle Ratios

Cutting them in half, you get two 3,4,5 triangles (pythagorean triple).

No need for Heron's Formula.

Height of 1st triangle(With area A)=Sqrt(5^2-3^2)=4

Height of 2nd triangle(With area B)=Sqrt(5^2-4^2)=3

Area A/ Area B = [(1/2)x6x4 ]/[(1/2)x8x3]= 1

in∆{5,5,6} call angle between side 5&6 : t so cos(t)=5^2+6^2-5^2(2×5×6)=.6 so sin(t)=.8 so area of ∆=1\2×5×6×.8=12, similarly in∆{5,5,8}. cos angle between 5&8=.8 so sin=.6 so area of∆=.5×5×8×.6=12. ratio =12\12=1####

Heros formula very easily...
S=a+b+c/2 s=8 s=9. put in it u ll get A=12 B=12 12/12= 1 Ans

1 its very simple................. AREA OF THE RED ONE(by Heron's Formula):- SIDES =5,5 and 6 and semi-perimeter=5+5+6/2=16/2=8 HERON'S FORMULA=√ s(s-a)(s-b)(s-c), where s=semi-perimeter, a,b,c are sides and so by substituting the values we get √8(8-5)(8-5)(8-6)=√8 3 3*2= √144=12(A)........ AND BY THE PROCESS WE CAN GET THE AREA OF THE BLUE TRIANGLE WHICH WILL BE ALSO = 12(B) SO A/B = 12/12=1 ANSWER....................

just go with phytagorean. The A got 4 wether the B got 3 in height. A= 0.5 x 6 x 4 = 12; B= 0.5 x 8 x 3 = 12; A/B=12/12=1.0

Herons formula is applicable in this question

for area A, find S= (a+b+c)/2= (5+5+6)/2=8 by using hero's formula AREA(A)=12 for area B, find S= (a+b+c)/2= (5+5+8)/2=9 AREA (B)= 12 => A//B= 1

4*6/2 = 12

3*8/2 = 12

12/12 = 1

heron's formula is also applicable directly

Here's another solution (without Heron's formula) Let A the base of the isoceles, and B be the another side. Cut each triangle into two. Then you can easily compute cos(t) for the oposite angle to the hypotenusa by $\cos{ (t) }=A/2B$

Using ${ (\sin{x}) }^{2} + { (\cos{x}) }^{2} = 1$ you can write $\sin{ (t) }= {( 1 - {A/2B}^{2} )}^{1/2}$

And the area of a triangle is $B*\sin{t}*A/2$

If you do this with each triangle you'll find that the ratio between the surfice of both triangles is actually 1

I though that the 5,8,8 triangle was bigger but maths never lie lol

I just felt both triangles have the same area so I answered 1 and got it right. :D This question feels like a typical optical illusion question where you are asked which is bigger. And the answer is always both objects are equal.

Using Herons Formula we see that both triangles have area $12$ therefore the answer is $\boxed1\;$ .