To determine whether a set of lengths can be the side lengths of a right triangle, we use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle with side lengths \( a \), \( b \), and hypotenuse \( c \) (where \( c \) is the longest side), the following equation holds:
\[ a^2 + b^2 = c^2 \]
Let's check each set of lengths:
1. **20 cm, 22 cm, 29 cm**
- Check: \( 20^2 + 22^2 = 400 + 484 = 884 \)
- Compare: \( 29^2 = 841 \)
- Since \( 400 + 484 \neq 841 \), these lengths do not form a right triangle.
2. **10 ft, 12 ft, 15 ft**
- Check: \( 10^2 + 12^2 = 100 + 144 = 244 \)
- Compare: \( 15^2 = 225 \)
- Since \( 100 + 144 \neq 225 \), these lengths do not form a right triangle.
3. **7 km, 24 km, 28 km**
- Check: \( 7^2 + 24^2 = 49 + 576 = 625 \)
- Compare: \( 28^2 = 784 \)
- Since \( 49 + 576 \neq 784 \), these lengths do not form a right triangle.
4. **13 in, 84 in, 85 in**
- Check: \( 13^2 + 84^2 = 169 + 7056 = 7225 \)
- Compare: \( 85^2 = 7225 \)
- Since \( 169 + 7056 = 7225 \), these lengths do form a right triangle.
From the calculations above, only the set of lengths \( 13 \) inches, \( 84 \) inches, and \( 85 \) inches can be the side lengths of a right triangle.
Which sets of lengths could be the side lengths of a right triangle
20cm,22cm,29cm
10ft,12ft,15ft
7km,24km,28km
13in,84in,85in
1 answer