Ok. EVERYONE! CHECK MY SITE AGAIN! CHECK THIS:
http://groups.msn.com/ROBERTschool/shoebox.msnw
IT SHOULD WORK! THE PICTURES ARE IN ORDER 1, 2, AND 3
A simpler way. If all the triangles and combined rectangles had their respective areas added. Triangle A (the top ) would have an area of 32 squares, and the bottom would be 33 squares. However if you solve for the area of the entire triangle , its 32.5. Therefore neither configuration is right on the correct cumulative area, The top triangle, if solved by adding the cumulative areas of the smaller triangles and rectangles is .5 square too small and the bottom one is actually .5 squares too big . Conclusion, its all done by squishing the y axis of the grid. No laws broken, its 3 card monty. It took longer to write a coherant answer than to solve
i know otherwise you couldn't see it
pshh...i dunno about you guys, but i could see that it wasn't 180 degrees...
maybe if you look REAL closely. the place where the blue and red triangle meet is where the line bends out..
But who could explain the square shape ?
when the pieces are rearranged, if drawn exactly correctly, the hypothenuse will have "shifted" diagonally upwards. the triangle from that amount of "shifting" is the missing area.
TechnoGuyRob wrote:
I hope you can see how this mystery triangle works!
No actually I don't, lol. I'm currently taking algebra 1...I have no idea what that formula meant. What does sqrt stand for??
Harrumph.
I contend that it's a matter of the demarcating lines.
In geometry, a "line" is a stream of points that has length but no width/thickness. A "line" has only ONE "dimension".
The "lines" in the composite figures shown have considerable width/thickness... they are TWO-dimensional.
The cumulative area of the lines ("gray area"!) is the disparity.
Solution
Here is a visual solultion:
It can also be solved fairly simply with basic trigonometry and some calculus (integrals). The solution is most understood by all visually, not everybody would be able to follow the math.
The Blue Triangle and the Red triangle are not similar. If I use the notation T(h,b) the red triangle is R(2,5) and the blue is B(3,8). If the triangles were similar there would be a common multiple (m) such that the blue triangle would be B(2m, 5m). This is not the case. As a result the slopes of the hypotenuse of the two triangles are not the same.
Rap
This has fooled a lot of people!!!!
I gave this to my 6th grade students and received a lot of creative answers, I also enjoyed how it fooled a lot of people on this board!!!!
Rap summed it up best though, the two triangles are not similar, therefore the hypotnuse of the quadrilateral that everyone thought was a triangle is not a straight line.
The madness was created by the yellow and green object's ? The green object's three grids length before decreasing to two. The yellow the opposite, to grids long before decreasing to three ? In the first example more or less a square in second form rectangle ? The area loss can be attributed to the diproportion between them ?
It is quite simple actually...the hypotenuse of the two triangles doesnt line up. they have two different angle measures. If you take a ruler/piece of paper (something straight) and hold it up to the hypotenuse of one triangle you will notice that it doesnt quite line up with the hypotenuse of the other triangle (see simple
)
Does the top hypotenuse have a hollow and the bottom hypotenuse a hump? i.e. the angles are dissimilar.
Two Riddles
I am having a tough time figuring these two out. Can anyone help?
#1: The thing you can hit without leaving a scar.
#2:Brown I am and much admired; Many horses I have tried; Tire a horse and worry a man; Tell me this riddle if you can.
Start a new topic and post this riddle there, Shadowghost.
is it just me, or are some just over thinking this(i've only looked over the first page of replies so maybe someone has already said this)
right off the bat, i noticed that if you start with the original triangle, at the upper right of where it starts, go the the left 5 squares, and down two, its obviously not the same ratio as the lower triangle, and is it just me, or is the top triangle's slant, not even a straight line, maybe i'm just tired and seeing things
