Batch 3 - Class 138 - Planar Geometry II

• In the "Crossing the Desert" problem, what if the truck also wants to come back?
• Answer: The pattern now is 1/2+1/4+1/6+...
• How far can the truck go? Answer: Again, the series is divergent and the truck can go any distance.

• (536Dudeney 294) When a bicycle is in motion, does the upper part of each wheel move faster than the bottom part near the ground? The instinctive response is no - since the wheel is a rigid whole, each part must move at the same pace. However, if you actually see a cycle going by, the top part does seem to move faster. Is it just an optical illusion?
• Answer: While every part of the wheel moves at the same pace around the axle, in a bicycle the axle is itself moving (which is what gives bicycle the motion). Since the upper part of the wheel moves in the same direction as the axle, it does move through the space faster than the bottom part, which is moving in opposite direction to the axle. To illustrate, draw a penny moving on a line, and trace the quarter journeys of top and bottom points. Kids will also notice that while the bottom part is rotating in the opposite direction to the direction of motion of the bicycle, it never really moves "backward".

• MartinShCol 5.15 - Five paper rectangles (one with a corner torn off) and six paper disks have been tossed on a table as shown below. Each visible corner of a rectangle and each spot where edges are seen to intersect marks a point. Find four sets of four "cocircular" points: four points that can be shown to lie on a circle.

• Answer: The bottom dark rectangle has all the four vertices visible, so those are cocircular. The white circle in the center has four points on the circle, so thats the second one. The other two are obtained by identifying "diagonals" of circles with two right angles suspended, which means those four form cocircular points. See red and white markers below.

• Slice the picture below into 2 sections and rearrange them to form a 6x4 rectangle

• Answer:      

  
• Slice the picture below into 2 sections and rearrange them to form a 8x8 square (without a hole in the middle)

• Answer: 

  
• MartinShCol 5.19 - Is the area of shaded region below less or more than 1/4 area of the circle?

• Answer: While the problem can be solved by long form math, and perhaps guessed visually, if one embeds the pattern of "bananas and deltas", the problem becomes obvious. The shaded area has 3 bananas and 1 delta, whereas the entire circle has 12 bananas and 6 deltas, so that shaded area is less than 1/4th of the circle.

• MartinShCol 5.5 - From one corner of a square, extend two lines that exactly trisect the square's area - Into what ratios do these lines cut the two sides of the square?

• Answer: 1/3 of each side. If you look at the diagonal originating from trisecting vertex, and look at one half of the square resulting from such a diagonal, the areas within that must be 1:2 ratio. However, the base of both those triangles is the same, so the height must be in 1:2 ratio.