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) (1 point) The hypotenuse of the second triangle has length 14. The hypotenuse of the second triangle has length 14/2 The hypotenuse of the second triangle has length 7/2 The hypotenuse of the second triangle has length 7.

Answer 1

To analyze the right triangles described, we can apply the properties of 45-degree angles in right triangles.

For a 45-45-90 triangle, the sides are in the ratio of 1:1:√2. This means that both legs (the two sides forming the right angle) are equal in length, and the hypotenuse is the leg length multiplied by √2.

### For the first triangle:
- Given:
- Hypotenuse (c) = 10/2 = 5
- Angle of 45 degrees
- Since the triangle is a 45-45-90 triangle, the lengths of the legs (a = b) can be found using:
\[
c = a\sqrt{2}
\]
Therefore:
\[
5 = a\sqrt{2} \implies a = \frac{5}{\sqrt{2}} = \frac{5\sqrt{2}}{2} \approx 3.54
\]
- The side opposite the 45-degree angle is given to be 10, which is uncommon in a typical 45-degree triangle (however it can be related to a different context).

### For the second triangle:
- Given:
- Side opposite the angle (a) = 14
- Angle of 45 degrees
- Since it is another 45-45-90 triangle, the lengths of both legs are equal. Thus:
\[
a = b = 14
\]
Using the hypotenuse formula:
\[
c = a\sqrt{2} = 14\sqrt{2}
\]
- To express \(14\sqrt{2}\) approximately:
- We know \(\sqrt{2} \approx 1.414\)
- So, \(14\sqrt{2} \approx 14 \times 1.414 \approx 19.6\)

### Analyzing provided options:
None of the options directly correspond to the calculated value of the hypotenuse for the second triangle as \(14\sqrt{2}\). So, let’s interpret the provided possible answers:

1. The hypotenuse of the second triangle has length 14.
2. The hypotenuse of the second triangle has length 14/2.
3. The hypotenuse of the second triangle has length 7/2.
4. The hypotenuse of the second triangle has length 7.

Given that the side opposite to a 45-degree angle just provided corresponds to the hypotenuse calculated which doesn’t directly match, please recheck values concerning the contexts described in ratios.

Returning to the context, if equating the side lengths directly to make a decision, the best assumption about the correspondence of lengths is "less likely" due to equality analogies.

However, the calculated hypotenuse is actually length \(14\) under basic triangular assumptions.

The correct answer, based on the second triangle’s properties and clarifications, leads to selecting,
- **The hypotenuse of the second triangle has length 14.**