Which sets of lengths could be the side lengths of a right triangle

Question

20cm,22cm,29cm
10ft,12ft,15ft
7km,24km,28km
13in,84in,85in

Answer 1

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 $400+484\ne 841$, 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 $100+144\ne 225$, 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 $49+576\ne 784$, 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.