To determine whether the frame is actually rectangular, we need to verify if the measured diagonal matches the expected diagonal for a rectangle with given dimensions. We can use the Pythagorean Theorem, which states that for a right-angled triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (length and width). Here's a step-by-step analysis:
1. Calculate the square of the length (14 inches) and the width (10 inches):
[tex]\[
14^2 = 196 \quad \text{and} \quad 10^2 = 100
\][/tex]
2. Sum these squares to find the expected square of the diagonal:
[tex]\[
196 + 100 = 296
\][/tex]
3. Calculate the actual square of the measured diagonal (20 inches):
[tex]\[
20^2 = 400
\][/tex]
4. Compare the expected square of the diagonal with the actual square of the measured diagonal:
- Expected square: [tex]\( 296 \)[/tex]
- Measured square: [tex]\( 400 \)[/tex]
Clearly, [tex]\( 296 \neq 400 \)[/tex]
5. Determine the actual length of the diagonal using the Pythagorean Theorem:
[tex]\[
\sqrt{296} \approx 17.2047
\][/tex]
Therefore, the calculated diagonal is approximately 17.2047 inches.
6. Check if the calculated diagonal (17.2047 inches) is close to the measured diagonal (20 inches):
Since [tex]\( 17.2047 \neq 20 \)[/tex], the calculated diagonal does not match the measured diagonal.
From these steps, we can conclude that the frame is not actually rectangular because the relationship given by the Pythagorean Theorem does not hold true.
The correct explanation is:
[tex]\[
\left(14^2 + 10^2\right) \neq 20^2
\][/tex]
Therefore, the true statement is:
[tex]\(
\left(14^2 + 10^2\right) \neq 20^2
\)[/tex]
