Question

teacher asks her students to design a game board for a class project. The dimensions of the boards created by four students are shown below.

Angela
length: 20 cm
width: 21 cm
diagonal: 29 cm
Bradley
length: 9 in.
width: 9 in.
diagonal: 9 in.
Carlton
length: 25 cm
width: 30 cm
diagonal: 35 cm
Della
length: 10 in.
width: 12 in.
diagonal: 15 in.

Whose game board could be a rectangle?
Angela
Bradley
Carlton
Della

Answer 1

To determine whose game board could be a rectangle, we can apply the Pythagorean theorem. For a rectangle with length \( l \), width \( w \), and diagonal \( d \), the relationship is given by:

\[
d^2 = l^2 + w^2
\]

We'll calculate \( d^2 \) for each student and compare it to \( l^2 + w^2 \) to check if the equality holds.

1. **Angela:**
- Length \( l = 20 \) cm, Width \( w = 21 \) cm, Diagonal \( d = 29 \) cm
- \( d^2 = 29^2 = 841 \)
- \( l^2 + w^2 = 20^2 + 21^2 = 400 + 441 = 841 \)
- **Equal**: Angela's board could be a rectangle.

2. **Bradley:**
- Length \( l = 9 \) in, Width \( w = 9 \) in, Diagonal \( d = 9 \) in
- \( d^2 = 9^2 = 81 \)
- \( l^2 + w^2 = 9^2 + 9^2 = 81 + 81 = 162 \)
- **Not equal**: Bradley's board cannot be a rectangle.

3. **Carlton:**
- Length \( l = 25 \) cm, Width \( w = 30 \) cm, Diagonal \( d = 35 \) cm
- \( d^2 = 35^2 = 1225 \)
- \( l^2 + w^2 = 25^2 + 30^2 = 625 + 900 = 1525 \)
- **Not equal**: Carlton's board cannot be a rectangle.

4. **Della:**
- Length \( l = 10 \) in, Width \( w = 12 \) in, Diagonal \( d = 15 \) in
- \( d^2 = 15^2 = 225 \)
- \( l^2 + w^2 = 10^2 + 12^2 = 100 + 144 = 244 \)
- **Not equal**: Della's board cannot be a rectangle.

### Conclusion:
The only student whose game board could be a rectangle is **Angela**.