Open Middle®
Directions: Using the digits 1 to 8, at most one time each, place a digit in each blank to create the vertices of a right triangle: A(__, __), B(__, __), C(__, __)
Extension: Try to do this using only the digits 1 to 6.
Hint
What are the properties of a right triangle?
Which lattice points can be used to create the triangle? Which lattice points can not be used?
Which lattice points can be used to create the triangle? Which lattice points can not be used?
Answer
Using the digits 1 to 8 there are a number of right triangles that can be created. Here are a few:
(1,3)(2,6)(8,4)
(1,7)(2,8)(4,6)
(1,7)(2,8)(6,4)
(1,6)(3,8)(4,7)
(1,3)(2,6)(8,4)
(1,7)(2,8)(4,6)
(1,7)(2,8)(6,4)
(1,6)(3,8)(4,7)
For each of the triangles above, an additional right triangle can be created by swapping the x and y coordinates (the reflection about line y=x).
Using the digits 1 to 6 there are less possibilities. Here is one:
(1,5)(3,2)(6,4)
and its reflection about the line y=x
(5,1)(2,3)(4,6)
Source: Erick Lee