The eight points highlighted in the figure divide the sides of the square into three equal parts. How many right triangles can be drawn with the three vertices at these points?

![](https://cdn.mathpix.com/cropped/2024_05_01_984603b876c0bc28095bg-12.jpg?height=337&width=328&top_left_y=1493&top_left_x=767)
A) 8
B) 12
C) 16
D) 24
E) 32

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Solution

## ALTERNATIVE D

Let's choose a point among the highlighted points; for example, the first point to the left on the bottom side of the square. Figure 1 shows the three right-angled triangles we can construct with the right angle vertex at this point. Since the same applies to the other highlighted points, we see that the number of right-angled triangles with vertices at these points is $8 \times 3=24$. We need to justify the statement that these triangles are right-angled. This is clear for the triangle in Figure 1.

As for the one in Figure 2, we note that the two white right-angled triangles are congruent, so their angles with the vertex at the chosen point sum to $90^{\circ}$, and consequently, the angle of the gray triangle at this vertex is also $90^{\circ}$.

Finally, the triangle in Figure 3 is right-angled because its shorter sides are diagonals of squares, as indicated by the lighter segments; thus, they form an angle of $45^{\circ}$ with the bottom side of the square, and the angle of the gray triangle at this vertex is also $90^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_984603b876c0bc28095bg-36.jpg?height=186&width=186&top_left_y=244&top_left_x=615)

Figure 1

![](https://cdn.mathpix.com/cropped/2024_05_01_984603b876c0bc28095bg-36.jpg?height=182&width=186&top_left_y=246&top_left_x=838)

Figure 2

![](https://cdn.mathpix.com/cropped/2024_05_01_984603b876c0bc28095bg-36.jpg?height=177&width=185&top_left_y=248&top_left_x=1067)

Figure 3

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