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Both Sharing Equal Area!
Which areas are into comparison?
Actually, areas of both are equal. The diagonal divide the rectangle into 2 halves. So triangle A and A' or B and B' have equal areas.
When diagonal divides the area of main rectangle into 2 halves, area of triangles A (or A') and area of triangle B (or B') are further subtracted from each half to get the areas of the shaded region.
Since equal areas are subtracted from triangles formed by diagonal to get the shaded area, the area of shaded parts are equal.
That is from each half area subtracted = A + B = A' + B'.
Divide The Cake Into Equal Parts!
I have just baked a rectangular cake when my wife comes home and barbarically cuts out a piece for herself. The piece she cuts is rectangular, but it’s not in any convenient proportion to the rest of the cake, and its sides aren’t even parallel to the cake’s sides.
I want to divide the remaining cake into two equal-sized halves with a single straight cut. How can I do it?
This is how it can be cut!
Cutting The Cake Into Equal Parts!
What was the problem?
Generally, a line drawn through the center of rectangle divides it into 2 equal parts.
Hence, a line drawn through the centers of both rectangles would divide each of them into 2 equal parts as shown below. (To get the center of each rectangle, all we need to do is draw diagonals of both).
Geometrical Puzzle - Solution
What was the puzzle?
Let's draw a line from each of vertex to the point at which all 4 regions intersect. This divides the given area into 2 triangles as shown below.
Obviously, here A and A' have equal area as they both share same base QS and height OT. Similarly, the areas of B & B', areas of C & C' and areas of D & D' must be equal.
Rewriting, A = A', B = B', C = C' and D = D'.
Now rewriting respective areas,
It's clear that,
A + B = 32
C + D = 16
Adding above 2 equations gives,
A + B + C + D = 32 + 16 = 48
But from figure, B + C = 20,
A + 20 + D = 48
A + D = 28.
That's the area of the shaded region which is equal to 28 Sq.cm
