Bot GPT 4-o mini In a right triangle with a 45-degree angle, the sides opposite the 45-degree angles are equal, and the relationships between the sides and the hypotenuse can be derived using the properties of 45-45-90 triangles.
The formula for the hypotenuse \(c\) in a 45-45-90 triangle is given by: \[ c = a\sqrt{2} \] where \(a\) is the length of each leg.
For the first triangle you mentioned, we have:
- Hypotenuse = \(10\sqrt{2}\)
- One side (opposite the 45-degree angle) = \(10\)
From the property, we can verify: \[ c = 10\sqrt{2} \]
Now, for the second triangle where the side opposite the 45-degree angle is \(14\): \[ c = 14\sqrt{2} \]
Since we need to express the hypotenuse in terms of the options provided, let’s evaluate these options:
- The hypotenuse of the second triangle has length \( \frac{14}{2} = 7\).
- The hypotenuse of the second triangle has length \( 14 \).
- The hypotenuse of the second triangle has length \( \frac{7}{2} = 3.5 \).
Clearly, the correct length of the hypotenuse for the second triangle is \( 14\sqrt{2} \), but we need to focus further using \(14\):
The choices given do not match \(14\sqrt{2}\), so none of the proposed options are accurate as they do not even simplify or provide a viable alternative directly equivalent. However, under the context of calculating the simple forms when asked for a comparison:
If \(c = 14\sqrt{2}\), let's simplify that for choices:
- \(14\) is not a correct hypotenuse
- \( \frac{14}{2} = 7 \) given
- or \(\frac{7}{2} = 3.5\)
The only acceptable value under rational assumption would be \(14\) which suffices as a significant number.
In conclusion, only the first two show that \(c\) can suffice as either whole or halved, but given that we follow law symmetrically it concludes the hypotenuse is: \[ \boxed{14} \text{ does not provide} \] Hence from angles directly evaluated, leading computation would be \(14\sqrt{2}\).
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