There is a point P inside a regular hexagon such that area of triangle

Question

There is a point P inside a regular hexagon such that area of  	riangle  PFA = 8 cm^2, area of  	riangle  PBC = 3 cm^2 and area of  	riangle  PDE = 5cm^2. Find the area of  	riangle PCD  (highlighted in green)?

A. 2 cm^2

B.  \frac{8}{3}  cm^2

C. 3 cm^2

D.  \frac{11}{3}  cm^2

E. 4 cm^2

yashikaaggarwal · Answer

Bunuel   chetan2u  Kindly solve this question.

chetan2u · Answer

An immediate answer. 
The opposite sides are parallel and distance between each set of parallel sides will be equal.

The height from P to each opposite parallel side would add up to H.

Note- On mobile, so will add a detailed explanation and sketch.

But the hexagon can be divided into equal areas by adding the areas of alternate triangles. 
PAB+PCD+PEF=PBC+PDE+PFA=8+5+3=16
So total area =16*2=32.

Now the opposite triangles should add up to 1/3 of total area as there are 3 sets of opposite triangle whose heights add up to same H and also the base is same. 
So each set of opposite triangles will have an area 32/3.
So area of highlighted triangle = 32/3-8=8/3

B

yashikaaggarwal · Answer

I didn't get the green bold highlight, can you explain it with diagram? Thank you

Lipun · Answer

h1 = 16/s, h2 = 10/s, h3 = 6/s
Height of equilateral triangle PQR = h1+h2+h3 = 32/s
Height of hexagon ABCDEF = (2/3)*Height of triangle PQR = 64/3s
=> h1 + h = 64/3s
=> h = 64/s - h1 = 64/3s - 16/s
=> h = 16/3s

Area = 1/2 * s * 16/3s = 8/3

Edit: Point inside hexagon is denoted by O.

nick1816 · Answer

Brilliant Bhai?

Posted from my mobile device

stne · Answer

Hi  Lipun , 
Can you please elaborate what relation you used to find the height of H1 , H2 and H3?

Lipun · Answer

Area of triangle PDE = 1/2*h2*s = 5 => h2 = 10/s. Similarly, for h1 and h3.
The height of an equilateral triangle is equal to the sum of the perpendicular distances of a point P from each of the sides.
Thus, H = h1+h2+h3 = 32/s

Hope this answers your query!

stne · Answer

Hi  Lipun  , 
Can you please check your figure, there are 2 points marked as P, causing some confusion in understanding your first explanation.
Thank you.

Lipun · Answer

Thanks  stne  for pointing out! Have edited to denote it as O.
Area of ODE = 1/2*h2*s = 5 => h2 = 10/s

Lipun

RTN · Answer

I still don't understand after going through the responses here, can someone please give another explanation?

There is a point P inside a regular hexagon such that area of triangle

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There is a point P inside a regular hexagon such that area of triangle

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