To determine which sets of numbers could be the sides of a right triangle, we can use the Pythagorean theorem, which states that for a right triangle with sides \( a \), \( b \), and hypotenuse \( c \):
\[ a^2 + b^2 = c^2 \]
We'll check each set of sides to see if it meets this criterion:
-
2.5, 6, and 6.5
- Check: \( 2.5^2 + 6^2 \) vs. \( 6.5^2 \)
- \( 2.5^2 = 6.25 \)
- \( 6^2 = 36 \)
- \( 6.5^2 = 42.25 \)
- \( 6.25 + 36 = 42.25 \) (valid)
-
2, 2, and 5
- Check: \( 2^2 + 2^2 \) vs. \( 5^2 \)
- \( 2^2 = 4 \)
- \( 5^2 = 25 \)
- \( 4 + 4 = 8 \) (not valid)
-
1.5, 2.5, and 4
- Check: \( 1.5^2 + 2.5^2 \) vs. \( 4^2 \)
- \( 1.5^2 = 2.25 \)
- \( 2.5^2 = 6.25 \)
- \( 4^2 = 16 \)
- \( 2.25 + 6.25 = 8.5 \) (not valid)
-
2, 3, and 5
- Check: \( 2^2 + 3^2 \) vs. \( 5^2 \)
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- \( 5^2 = 25 \)
- \( 4 + 9 = 13 \) (not valid)
From the options provided, the only valid set that could form a right triangle is 2.5, 6, and 6.5.