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

Chameleons

On an island live 13 purple, 15 yellow and 17 maroon chameleons. When two chameleons of different colors meet, they both change into the third color. Is there a sequence of pairwise meetings after which all chameleons have the same color?

 

Let <p, y, m> denote a population of p purple, y yellow and m maroon chameleons. Can population <13, 15, 17> can be transformed into <45, 0, 0> or <0, 45, 0> or <0, 0, 45> through a series of pairwise meetings? Define function X(p, y, m) = (0p + 1y + 2m) mod 3. An interesting property of X is that its value does not change after any pairwise meeting because X(p, y, m) = X(p-1, y-1, m+2) = X(p-1, y+2, m-1) = X(p+2, y-1, m-1). Now X(13, 15, 17) equals 1. However, X(45, 0, 0) = X(0, 45, 0) = X(0, 0, 45) = 0. This means that there is no sequence of pairwise meetings after which all chameleons will have identical color.

Scales

Review the following picture of scale balances:

Based on the balances in the above picture, which of the following balances can be correct?

Answer is C.