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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