Larger, Smaller or Similar?

Which of the yellow areas is larger?

Here is comparison of areas! 

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'.

Geometrical Puzzle

Find the area of the shaded region.

Escape to the answer! 


Source 

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

"What's The Area of The Triangle?"

If I place a 6 cm × 6 cm square on a triangle, I can cover up to 60% of the triangle. If I place the triangle on the square, I can cover up to 2/3 of the square. What is the area, in cm², of the triangle?

(a) 22 4/5
(b) 24
(c) 36
(d) 40
(e) 60

Here is that mathematical calculation!

Calculation of Area of Triangle

What was the question?

The most important thing to note here is the area that square overlaps on the triangle is equal to the area that triangle overlaps the square with the maximum contact area. That is both the areas are supposed to be equal in amount. 


So if T is the area of the triangle & S is area of the square then,

0.6 T = 2/3  x S = 2/3 x (6x6) = 2/3 x 36 = 24

T = 40 Square cm.

Hence, answer is option (d) 40.