Exercise 27 Page 600

Examining the points, we notice that we have not been given any numerical coordinates. Rather, we are given the coordinates of $△NOP$ relative to those of $△ABC.$ To visualize the triangles, we will assign values to the vertice's coordinates of $△ABC.$ We will choose our values to make the calculations of $△NO{P}^{\prime }s$ vertices as simple as possible.

As we can see, $u=0,$ $v=0,$ $w=1,$ $x=1,$ $y=0,$ and $z=1.$ Having assigned these values, we can determine the coordinates of $△NOP.$

Now we can draw $△NOP$ in the same coordinate plane as $△ABC.$ Notice that both of these triangles are right triangles. Let's mark this as well.

If the triangles are similar they have to have the same shape. Examining the diagram, we see that $\overline{AC}$ and $\overline{NP}$ are both vertical sides drawn along the $y\text{-}$axis. Therefore, if we reflect $△ABC$ in the $x\text{-}$axis, these sides will map onto each other. If $\overline{AB}$ also maps onto $\overline{NO},$ the triangles have the same shape.

Since $\overline{AB}$ does not map onto $\overline{NO},$ the triangles do not have the same shape. Therefore, they are not similar.