Sander Parallelogram

Sander Parallelogram 

Sander Parallelogram

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Note:The two lines in purple have the same length; Sander's Parallelogram, also known as "Sander Illusion" is an optical illusion described by the German psychologist Friedrich Sander.

The Ehrenstein Illusion

The Ehrenstein Illusion

The Ehrenstein Illusion is depicted below. As shown in the following picture, the straight sides of the square appear curved due to the concentric circles.

The Ehrenstein Illusion

The Penrose Triangle

The Penrose Triangle

The Penrose Triangle, also known as The Penrose Tribar, The Tribar and The Impossible Tribar is an impossible object created by a Swedish artist, Oscar Reutersvärd, in 1934. The Penrose Triangle is depicted below:

The Penrose Triangle

 

Finding Isosceles Triangles

Mr. Isocel asked students in his class to find isosceles triangles whose vertices lie on a coordinate grid. For each student example below, explain why the triangle is isosceles.

Monika draws the following triangle:

 

Karl's picture is here:

Bob draws this picture:

Now, you find a general solution (a generic method)!

 





Answer:

1. We can see that Monika's triangle is isosceles in several ways. We will show that $|AC| = |BC|$. We can use the Pythagorean Theorem to check that these both have length $\sqrt{2^2 + 3^2}$. In the calculations below $M = (3,1)$ is the midpoint of $\overline{AB}$.
   
   
   $$\begin{align} |AC|^2 &= |AM|^2 + |MC|^2 = 2^2 + 3^2\\ |BC|^2 &= |BM|^2 + |MC|^2 = 2^2 + 3^2. \end{align}$$
   We can also observe that to get from $A$ to $C$ we go over (to the right) by two units and then up three units. To get from $B$ to $C$ we go over (to the left) by two units and then up three units. We travel the same distance to get from $A$ or $B$ to $C$ so $A$ and $B$ are equidistant from $C$. Finally, we can draw the horizontal line of symmetry for △ABC.
   
   
   
   Isosceles triangles are exactly those triangles with a line of symmetry so $\triangle ABC$must be isosceles.

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Next Domino

What would be the next domino in the following sequence?

m (+5)            >>> r
68 - (5) * 4      >>> 48
r - 48 == 18 - 48 >>> -30