Mensa Brain Twister Geometry Solutions:

 1) The Rectangle Problem:

The height of a rectangle is 81 units. Two circles are inscribed in the rectangle as shown above. The circles are tangent to each other. A line segment has an endpoint on both circles and passes through the point of tangency. It measures 56 units, and is parallel to the height of the rectangle. What is the length L of the rectangle?

Solution:

Reconstruct to obtain:

Then let R be the radius of the larger circle and r be the radius of the smaller one.

Let x be half the chord in the larger circle and y be half the chord in the smaller,

Then from the original dimensions assigned:  

2x + 2y = 56   ®  x + y = 28 (equals one leg of the yellow triangle)

Let z be the other leg of the yellow triangle. Then:

R = x + y + r = 81

R + 28 + r = 81    ®  R + r = 53 (equals the hypote4neous of yellow triangle)

Then by Pythagoras' theorem:  

28 ²  +  z²  =  53²   ®   z²    =   53²   - 28 ²   = 2025

So:  z =  (2025) ^1/2  =   45

Then the length of the rectangle is:

R + z + r =  53 + 45 = 98

1) The Square Problem:   

ABCD is a square. W is the midpoint of AB.  Z is the center of the square where AD and BC intersect. X is the point where AD and WC intersect. Y is the point where BC and WD intersect. 

If the area of square ABCD is 4 square units, then find the area of the orange region WXYZ.   y= -x

Solution:

AWZ  is one half of one fourth of the square. (I.e. One half of triangle ABZ The other fourths are the triangles:  ACZ, DCZ and BDZ)   Therefore the area of AWZ =   (1/2) x (1/4)  x 4 =  1/2

The  area of WXYZ = 2x the area  of  WZX where

WZX  = AWZ -  AWX  = ®  -  AVX  -  VWX

Now, if A is the origin of the coordinate plane:

The slope of AX is -1    ®  y  = - x

The slope of XW is  2     ®  y  = 2x  - 2 

Then AX and XW intersect where:  -x =  2x - 2

Then: 3x = 2  ®  x = 2/3

The area of AVX = (2/3) (2/3) (1/2) =  4/18

The area of VWX = (1/3) (2/3) (1/2) =  2/18

The area of WXYZ  = 2 [ 9/18 - 6/18]  = 2 [3/18] = 6/18 = 1/3