Find Area of Quadrilateral that is shaded in Pink

This question is uploaded on Instagram on page "@gercekboss" on 1st of may 2020. I provided Instagram link of this question. Click here to go to that question.

Take a look at figure given below. Give it a try and when you are ready then watch the solution.

As you can see here we have to find area of pink region which is surrounded by some different colored region with given area. In this type of question most of the time similarity we can get answer by finding similar triangles. In this question also I will find similar triangles but first I will label this diagram for making calculations easier. And for few steps we don't need given areas of 32, 3, 8, 27 square units.

Now take a look at Green and Yellow triangle as well as Blackish Yellow and Blue triangle as given below. It's one angle is same by vertically opposite angles and other two lines are in same ratio. For Green pair of triangles, angles forming at F are vertically opposite angles and for pink pair of triangles, angle forming at H are vertically opposite angles. As well as ratio of AF and FG and ratio of EF and  FB are same and also ratio of EH and HC and ratio of DH and HG are same.

By this relation triangles AFE and GFB are similar as well as triangles EHD and CHG are similar.

Now I draw some vertical and horizontal lines as shown in figure below. And I also marked b1 and h1 as base and height of 1st triangle similarly for 2nd triangle b2 and h2 for 3rd triangle b3 and h3 and for 4th triangle b4 and h4.

We know that for similar triangles Height:Base ratio is always same.

Consider Green and Yellow triangle.

h1/b1 = h2/b2

⇒ h1/h2 = b1/b2 ---------(1)

Now consider areas of Green and Yellow Triangles,

(1/2 × b1 × h1)/(1/2 × b2 × h2) = 32/8

⇒ (b1 × h1)/(b2 × h2) = 4/1

⇒ b1^2/b2^2 = h1^2/h2^2 = 4/1 ---------(From 1)

⇒ b1/b2 = h1/h2 = 2/1

Now consider Blackish Yellow and Blue triangle.

h3/b3 = h4/b4

⇒ h3/h4 = b3/b4 ----------(2)

Now consider areas of Blackish Yellow and Blue Triangles,

(1/2 × b3 × h3)/(1/2 × b4 × h4) = 3/27

⇒ (b3 × h3)/(b4 × h4) = 1/9

⇒ b3^2/b4^2 = h3^2/h4^2 = 1/9 ---------(From 1)

⇒ b3/b4 = h3/h4 = 1/3

Let,

h1 = 2p

h2 = p

b1 = 2q

b2 = q

h3 = r

h4 = 3r

b3 = s

b4 = 3s

As we know opposite sides of Rectangle are same

⇒ 2p+p = r+3r

⇒ 3p = 4r

⇒ r = 3p/4

also,

⇒ 2q+s = q+3s

⇒ q = 2s

Now convert all q and r in terms of p and s respectively as in below figure.

As we know Yellow area is 8 nut as of above figure,

Yellow Area = 1/2 × p × 2s

⇒ 8 = ps

Now just consider Yellow, Pink and Blue triangle. They combine to form a large triangle with Height and Base are as of Rectangle.

Now area of Large triangle = Yellow area + Pink area + Blue area

1/2 × Base × Height = 8 + Pink area + 27

1/2 × 5s × 3p = 35 + Pink area

15/2 × ps = 35 + Pink area

15/2 × 8 = 35 + Pink area ------(ps = 8)

60 = 35 + Pink area

Pink area = 25 square units