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 3. 8; 29; 11; 12
 4. B. 11, 5, 13
 5. D. sides are 1; hypotenuse is √2
Step-by-step explanation:
3. Each of these can be figured from the Pythagorean theorem. The right-most blank is presumed to be the hypotenuse, so for all but the second blank the missing number is the root of the difference of the squares. Interestingly, you can compute this in your head by using the factoring of the difference of squares:
 a = √(c^2 -b^2) . . . . . . where c is the hypotenuse
 a = √((c +b)(c -b)) . . . . difference of squares is factored
For the first, this is b = √((10+6)(10 -6)) = √(16·4) = 4·2 = 8
For the third, this is a = √((61+60)(61-60)) = √121 = 11
For the fourth, this is b = √((13+5)(13-5)) = √(18·8) = √(9·2·2·4) = 3·2·2 = 12
For the second blank, we can do it the usual way:
 c = √(20² +21²) = √(400 +441) = √841 = 29
The blanks are filled with 8, 29, 11, 12.
Note: You may notice that (6, 8, 10) is double the triple (3, 4, 5). The triple (5, 12, 13) comes into play in the next problem.
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4. The first few Pythagorean triples are (3, 4, 5), (5, 12, 13), (7, 24, 25). It can be useful to remember these.
By comparing these to the first three choices, you can see that choice B lists three numbers that cannot be a right triangle. (If the shortest and longest sides are 5 and 13, then the other side must be 12 if it is a right triangle.)
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5. Darnell's finding that 1² +1² = (√2)² means that a triangle with sides in the ratio 1 : 1 : √2 is a right triangle. This description matches choice D.
Note: It can be useful to remember that sides 1 : 1 : √2 make an isosceles right triangle. It means the diagonal of a square is √2 times the side length, for example.
