To determine whether the given sets of measurements can represent the lengths of the sides of a right triangle, we can apply the Pythagorean theorem. According to this theorem, for three sides \( a \), \( b \), and \( c \) of a triangle (where \( c \) is the longest side), the following must hold true:
\[ c^2 = a^2 + b^2 \]
Let's evaluate each set:
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Set: 4, 5, 6
- Sort the sides: \( a = 4 \), \( b = 5 \), \( c = 6 \) (the longest side).
- Check if \( 6^2 = 4^2 + 5^2 \):
- \( 6^2 = 36 \)
- \( 4^2 + 5^2 = 16 + 25 = 41 \)
- Since \( 36 \neq 41 \), this set does not form a right triangle.
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Set: 29, 20, 21
- Sort the sides: \( a = 20 \), \( b = 21 \), \( c = 29 \).
- Check if \( 29^2 = 20^2 + 21^2 \):
- \( 29^2 = 841 \)
- \( 20^2 + 21^2 = 400 + 441 = 841 \)
- Since \( 841 = 841 \), this set does form a right triangle.
-
Set: 63, 16, 65
- Sort the sides: \( a = 16 \), \( b = 63 \), \( c = 65 \).
- Check if \( 65^2 = 16^2 + 63^2 \):
- \( 65^2 = 4225 \)
- \( 16^2 + 63^2 = 256 + 3969 = 4225 \)
- Since \( 4225 = 4225 \), this set does form a right triangle.
Summary:
-
Can form a right triangle:
- 29, 20, 21
- 63, 16, 65
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Cannot form a right triangle:
- 4, 5, 6