Question
Nadine is making a quilt. She can use a rectangle of any length and width, as long as the angles in the shape are right angles. Which pieces of fabric can Nadine use in the quilt? Check all that apply.
Fabric 1
length: 5 cm
width: 12 cm
diagonal: 13 cm
Fabric 2
length: 8 cm
width: 6 cm
diagonal: 10 cm
Fabric 3
length: 4 cm
width: 5 cm
diagonal: 6 cm
Fabric 4
length: 10 cm
width: 8 cm
diagonal: 12 cm
Fabric 5
length: 6 cm
width: 4 cm
diagonal: 8 cm
Fabric 6
length: 8 cm
width: 15 cm
diagonal: 17 cm
Fabric 1
Fabric 2
Fabric 3
Fabric 4
Fabric 5
Fabric 6
Fabric 1
length: 5 cm
width: 12 cm
diagonal: 13 cm
Fabric 2
length: 8 cm
width: 6 cm
diagonal: 10 cm
Fabric 3
length: 4 cm
width: 5 cm
diagonal: 6 cm
Fabric 4
length: 10 cm
width: 8 cm
diagonal: 12 cm
Fabric 5
length: 6 cm
width: 4 cm
diagonal: 8 cm
Fabric 6
length: 8 cm
width: 15 cm
diagonal: 17 cm
Fabric 1
Fabric 2
Fabric 3
Fabric 4
Fabric 5
Fabric 6
Answers
GPT-4o mini
To determine which pieces of fabric can be used for a quilt, we need to verify that the diagonal of each rectangle follows the Pythagorean theorem. The theorem states that for a right-angled triangle, the square of the length of the hypotenuse (d, the diagonal) is equal to the sum of the squares of the lengths of the other two sides (l and w, the length and width):
\[ d^2 = l^2 + w^2 \]
Let's calculate for each fabric piece:
**Fabric 1:**
- Length (l) = 5 cm
- Width (w) = 12 cm
- Diagonal (d) = 13 cm
Check:
\[ 13^2 = 5^2 + 12^2 \]
\[ 169 = 25 + 144 \]
\[ 169 = 169 \] (True)
**Fabric 2:**
- Length (l) = 8 cm
- Width (w) = 6 cm
- Diagonal (d) = 10 cm
Check:
\[ 10^2 = 8^2 + 6^2 \]
\[ 100 = 64 + 36 \]
\[ 100 = 100 \] (True)
**Fabric 3:**
- Length (l) = 4 cm
- Width (w) = 5 cm
- Diagonal (d) = 6 cm
Check:
\[ 6^2 = 4^2 + 5^2 \]
\[ 36 = 16 + 25 \]
\[ 36 = 41 \] (False)
**Fabric 4:**
- Length (l) = 10 cm
- Width (w) = 8 cm
- Diagonal (d) = 12 cm
Check:
\[ 12^2 = 10^2 + 8^2 \]
\[ 144 = 100 + 64 \]
\[ 144 = 164 \] (False)
**Fabric 5:**
- Length (l) = 6 cm
- Width (w) = 4 cm
- Diagonal (d) = 8 cm
Check:
\[ 8^2 = 6^2 + 4^2 \]
\[ 64 = 36 + 16 \]
\[ 64 = 52 \] (False)
**Fabric 6:**
- Length (l) = 8 cm
- Width (w) = 15 cm
- Diagonal (d) = 17 cm
Check:
\[ 17^2 = 8^2 + 15^2 \]
\[ 289 = 64 + 225 \]
\[ 289 = 289 \] (True)
Now we can summarize the results:
- **Fabric 1**: Valid
- **Fabric 2**: Valid
- **Fabric 3**: Invalid
- **Fabric 4**: Invalid
- **Fabric 5**: Invalid
- **Fabric 6**: Valid
Therefore, the pieces of fabric that Nadine can use are:
- **Fabric 1**
- **Fabric 2**
- **Fabric 6**
\[ d^2 = l^2 + w^2 \]
Let's calculate for each fabric piece:
**Fabric 1:**
- Length (l) = 5 cm
- Width (w) = 12 cm
- Diagonal (d) = 13 cm
Check:
\[ 13^2 = 5^2 + 12^2 \]
\[ 169 = 25 + 144 \]
\[ 169 = 169 \] (True)
**Fabric 2:**
- Length (l) = 8 cm
- Width (w) = 6 cm
- Diagonal (d) = 10 cm
Check:
\[ 10^2 = 8^2 + 6^2 \]
\[ 100 = 64 + 36 \]
\[ 100 = 100 \] (True)
**Fabric 3:**
- Length (l) = 4 cm
- Width (w) = 5 cm
- Diagonal (d) = 6 cm
Check:
\[ 6^2 = 4^2 + 5^2 \]
\[ 36 = 16 + 25 \]
\[ 36 = 41 \] (False)
**Fabric 4:**
- Length (l) = 10 cm
- Width (w) = 8 cm
- Diagonal (d) = 12 cm
Check:
\[ 12^2 = 10^2 + 8^2 \]
\[ 144 = 100 + 64 \]
\[ 144 = 164 \] (False)
**Fabric 5:**
- Length (l) = 6 cm
- Width (w) = 4 cm
- Diagonal (d) = 8 cm
Check:
\[ 8^2 = 6^2 + 4^2 \]
\[ 64 = 36 + 16 \]
\[ 64 = 52 \] (False)
**Fabric 6:**
- Length (l) = 8 cm
- Width (w) = 15 cm
- Diagonal (d) = 17 cm
Check:
\[ 17^2 = 8^2 + 15^2 \]
\[ 289 = 64 + 225 \]
\[ 289 = 289 \] (True)
Now we can summarize the results:
- **Fabric 1**: Valid
- **Fabric 2**: Valid
- **Fabric 3**: Invalid
- **Fabric 4**: Invalid
- **Fabric 5**: Invalid
- **Fabric 6**: Valid
Therefore, the pieces of fabric that Nadine can use are:
- **Fabric 1**
- **Fabric 2**
- **Fabric 6**