Question

A) A right triangle has a hypotenuse of length 10√2 and an angle of 45 degrees, with a side opposite this angle with a length of 10. A second right triangle also has an angle of 45 degrees, with a side opposite this angle with a length of 14. Determine the length of the hypotenuse in the second triangle. (1 point) The hypotenuse of the second triangle has length 14/2 The hypotenuse of the second triangle has length 14. The hypotenuse of the second triangle has length 7/2

Answer 1

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:

1. The hypotenuse of the second triangle has length \( \frac{14}{2} = 7\).
2. The hypotenuse of the second triangle has length \( 14 \).
3. 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 \(14\sqrt{2}\).