This is for a geometry test i'm studying for,

Check the picture below.

so, we know this is a square, so all sides are the same length, but that simply also means that the altitude or height of the inscribed triangle is 36, as well as its base.

so, if we take the area of the square, that will include the area of the triangle, of course that's simply 36*36.

however, if we also take the area of the inscribed triangle and then subtract that much from the area of the square, we're in effect making a triangular hole in the square, and what's leftover, is the shaded area.

56.6

step-by-step explanation:

the ratio of the perimeters is equal to the scale.

the ratio of the areas is equal to the square of the scale.

34 / 20 = k

a / 19.6 = k²

substituting:

a / 19.6 = (34 / 20)²

a = 19.6 (34 / 20)²

a ≈ 56.6

let's call the solutions x₁ = (5 - 2√7)/3 and x₂ = (5 + 2√7)/3

then the equation is (x-x₁)(x-x₂) = 0

if you fill that in you get a lot of numbers which you can recognize as ax²+bx+c=0.

step-by-step explanation:

you can see that a=1 (there is just an x²)

b = (-5 -2√7 + 2√7 - 5)/3 = -10/3

c = -2√7)/3 ) * (-5+2√7)/3) = -1/3

so your answer is 3x^2 – 10x – 1 = 0