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# Pythagorean Theorem 2

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.

### Hint

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 the leg long enough and only two digits. It also can’t be 4 or less as the hypotenuse has to be the longest side.

So, the possible side lengths for the triangles are (a, b, and c being the leg, leg, and hypotenuse):

a, b, c
4, √84, 10
4, √65, 09
4, √48, 08
4, √33, 07
4, √20, 06
4, 03, 05

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

a, b, c
4, √65, 09 (from Karl S.)
4, √84, 10

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

## Minimize Slope

Directions: Given the point (3,5), use digits 1-9, at most one time, to find a …