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. The theorem states that for a right triangle with sides \( a \) and \( b \), and hypotenuse \( c \), the relationship \( a^2 + b^2 = c^2 \) must hold true, where \( c \) is the longest side.
Let's analyze each set of measurements:
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63 in, 16 in, 65 in
- Identify the longest side: \( c = 65 \) in
- Check if \( 63^2 + 16^2 = 65^2 \): \[ 63^2 = 3969 \quad \text{and} \quad 16^2 = 256 \quad \Rightarrow \quad 63^2 + 16^2 = 3969 + 256 = 4225 \] \[ 65^2 = 4225 \]
- Since \( 63^2 + 16^2 = 65^2 \), this set of measurements does describe the sides of a right triangle.
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4 m, 5 m, 6 m
- Identify the longest side: \( c = 6 \) m
- Check if \( 4^2 + 5^2 = 6^2 \): \[ 4^2 = 16 \quad \text{and} \quad 5^2 = 25 \quad \Rightarrow \quad 4^2 + 5^2 = 16 + 25 = 41 \] \[ 6^2 = 36 \]
- Since \( 4^2 + 5^2 \neq 6^2 \), this set of measurements does not describe the sides of a right triangle.
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29 in, 20 in, 21 in
- Identify the longest side: \( c = 29 \) in
- Check if \( 20^2 + 21^2 = 29^2 \): \[ 20^2 = 400 \quad \text{and} \quad 21^2 = 441 \quad \Rightarrow \quad 20^2 + 21^2 = 400 + 441 = 841 \] \[ 29^2 = 841 \]
- Since \( 20^2 + 21^2 = 29^2 \), this set of measurements does describe the sides of a right triangle.
Summary
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Right Triangle:
- 63 in, 16 in, 65 in
- 29 in, 20 in, 21 in
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Not a Right Triangle:
- 4 m, 5 m, 6 m
1 answer