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The blue region has area 15 cm².

Find the area of the brown region.
Show the working out and formula.

The blue region has area 15 cm². Find the area of the brown region. Show the working-example-1
User LondonGuy
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1 Answer

5 votes

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

User Jer
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