Explanation:
let's start with some areas we know about, or that we can define by a variable.
the area of the main rectangle (orange area and larger blank triangle) is
10×8 = 80cm²
the area of the larger blank triangle is (it is a right-angled triangle)
10×x/2 = 5x cm²
with x being the part of the 8 cm side that is the shorter leg of the larger blank right-angled triangle.
the area of the smaller blank right-angled triangle is
(8 - x) × y / 2 = (8y - xy)/2
with y being the longer leg of that right-angled triangle.
and then we have the large right-angled triangle on the right side of the whole shape (and it contains the blue area) :
8 × y / 2 = 4y cm²
and that is the sum of the smaller blank triangle and the blue area :
4y = 15 + (8y - xy)/2
8y = 30 + 8y - xy
0 = 30 - xy
xy = 30
y = 30/x
now, let's imagine the large right-angled triangle containing of the orange area and the smaller blank triangle to be a dilation of the smaller blank triangle from the bottom right vertex all the way to the left 8 cm side.
so, in this dilation, the left orange area side (8 cm) is the right orange area side (8 - x cm) times a factor f :
8 = (8 - x)×f = 8f - xf
the ground lines must follow the same scale (ratio) and therefore the same factor :
(10 + y) = y×f
when we use the identity above (y = 30/x), we get
10 + 30/x = 30f/x
10x + 30 = 30f
x + 3 = 3f
x = 3f - 3
we use that in the previous equation
8 = 8f - xf = 8f - (3f - 3)f = 8f - 3f² + 3f = 11f - 3f²
that means
3f² - 11f + 8 = 0
for a quadratic equation
ax² + bx + c = 0
the general solution is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
a = 3
b = -11
c = 8
x = f
f = (11 ± sqrt((-11)² - 4×3×8))/(2×3) =
= (11 ± sqrt(121 - 96))/6 = (11 ± sqrt(25))/6 =
= (11 ± 5)/6
f1 = (11 + 5)/6 = 16/6 = 8/3
f2 = (11 - 5)/6 = 6/6 = 1
f = 1 would make x = 0, and we would get for y = 30/0, which is undefined.
so, f = 8/3 is our valid solution for the scaling factor. that gives us then for x and y :
x = 3f - 3 = 3×8/3 - 3 = 8 - 3 = 5 cm
xy = 30
5y = 30
y = 30/5 = 6 cm
that makes the area of the larger blank triangle
5x = 5×5 = 25 cm²
the orange area is the rectangle (80 cm²) minus the larger blank triangle
orange area = 80 - 25 = 55 cm²