Question
Tasha assembled a picture frame that is advertised as rectangular. The completed frame is 14 inches long and 10 inches wide. She measured the diagonal length across the frame as 20 inches. Which best explains why the frame cannot actually be rectangular?
14 squared + 10 squared not-equals 20 squared
14 + 10 not-equals 20
(14 + 10) squared not-equals 20 squared
(14 squared + 10 squared) squared not-equals 20 squared
14 squared + 10 squared not-equals 20 squared
14 + 10 not-equals 20
(14 + 10) squared not-equals 20 squared
(14 squared + 10 squared) squared not-equals 20 squared
Answers
GPT-4o mini
To determine if the frame can be a rectangle, we can use the Pythagorean theorem, which states that for a rectangle, the square of the diagonal (d^2) should equal the sum of the squares of the lengths of the sides (a^2 + b^2).
In Tasha's case:
- Length (a) = 14 inches
- Width (b) = 10 inches
- Diagonal (d) = 20 inches
According to the Pythagorean theorem, we check:
\[ d^2 = a^2 + b^2 \]
Calculating each term:
- \( a^2 = 14^2 = 196 \)
- \( b^2 = 10^2 = 100 \)
- Therefore, \( a^2 + b^2 = 196 + 100 = 296 \)
Now calculating \( d^2 \):
- \( d^2 = 20^2 = 400 \)
Now, comparing the two:
\[ 400 \neq 296 \]
Thus, since \( 14^2 + 10^2 \) does not equal \( 20^2 \), the statement that best explains why the frame cannot be rectangular is:
**14 squared + 10 squared not-equals 20 squared.**
In Tasha's case:
- Length (a) = 14 inches
- Width (b) = 10 inches
- Diagonal (d) = 20 inches
According to the Pythagorean theorem, we check:
\[ d^2 = a^2 + b^2 \]
Calculating each term:
- \( a^2 = 14^2 = 196 \)
- \( b^2 = 10^2 = 100 \)
- Therefore, \( a^2 + b^2 = 196 + 100 = 296 \)
Now calculating \( d^2 \):
- \( d^2 = 20^2 = 400 \)
Now, comparing the two:
\[ 400 \neq 296 \]
Thus, since \( 14^2 + 10^2 \) does not equal \( 20^2 \), the statement that best explains why the frame cannot be rectangular is:
**14 squared + 10 squared not-equals 20 squared.**