Question

Drag and drop the given set of measurements into the correct box to describe whether the measurements can describe the lengths of the three sides of a right triangle or not. Show your work.(3 points)

4,5,6
29,20,21
63,16,65

Answer 1

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:

1. **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**.

2. **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**.

3. **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

- **Cannot form a right triangle:**
- 4, 5, 6