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.
63 in, 16 in, 65 in
4 m, 5 m, 6 m
29 in, 20 in, 21 in

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. 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:

1. **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.

2. **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.

3. **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
- **Right Triangle**:
- 63 in, 16 in, 65 in
- 29 in, 20 in, 21 in

- **Not a Right Triangle**:
- 4 m, 5 m, 6 m