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Pythagorean Theorem

Directions: Using the digits 0 to 9, at most one time each, fill in the boxes to find the lengths of the missing sides such that the missing leg’s length is as long as possible.



How do we figure out which side length would be easiest to assume?
How do we figure out the length of the remaining side?
Which side of a right triangle is the longest?



The longest possible length for the hypotenuse is 10. If it was 11, you would not be able to make a leg long enough with only two digits. It also can’t be 4 or less as it would b

So, the possible side lengths for the triangles are:

a, b, c
4, √84, 10
4, √48, 8
4, √33, 7
4, √20, 6
4, 3, 5

After removing the solutions that duplicate digits, you are left with two options:

a, b, c
4, √84, 10
4, 3, 5

Therefore, 4, √84, 10 is the optimal solution. Be on the lookout though for students who make the leg longer than the hypotenuse. While we want the missing leg to be as long as possible, it still has to be less than the length of the hypotenuse. So, this is a good chance to check for misunderstandings.

Source: Robert Kaplinsky

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  1. Love it!

  2. 4 and 84 share a digit of 4. Shouldn’t that response be eliminated, too?

  3. Couldn’t 4, sqrt(65), and 09 also be a possible solution, albeit not as long as your given optimal solution?

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