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Original Question
A right triangle has a hypotenuse of length 16 and an angle of 45°, with a side opposite this angle of length 82–√. A second ri...Question
A right triangle has a hypotenuse of length 16 and an angle of 45°, with a side opposite this angle of length 82–√
. A second right triangle also has an angle of 45° and a side opposite this angle with a length of 42–√
. Determine the length of the hypotenuse in the second triangle.(1 point)
Responses
The hypotenuse of the second triangle has length 42–√
.
The hypotenuse of the second triangle has length 4 Start Root 2 End Root .
The hypotenuse of the second triangle has length 4.
The hypotenuse of the second triangle has length 4.
The hypotenuse of the second triangle has length 8.
The hypotenuse of the second triangle has length 8.
The hypotenuse of the second triangle has length 82–√
.
. A second right triangle also has an angle of 45° and a side opposite this angle with a length of 42–√
. Determine the length of the hypotenuse in the second triangle.(1 point)
Responses
The hypotenuse of the second triangle has length 42–√
.
The hypotenuse of the second triangle has length 4 Start Root 2 End Root .
The hypotenuse of the second triangle has length 4.
The hypotenuse of the second triangle has length 4.
The hypotenuse of the second triangle has length 8.
The hypotenuse of the second triangle has length 8.
The hypotenuse of the second triangle has length 82–√
.
Answers
GPT-4o mini
In a right triangle with angles of 45°, the sides opposite the 45° angles are equal, and the relationship between the length of a side opposite a 45° angle (let's call this side \( x \)) and the hypotenuse (\( h \)) is given by:
\[
h = x \sqrt{2}
\]
Now, apply this to the second triangle, where the side opposite the 45° angle is \( 42 - \sqrt{2} \).
Thus, the hypotenuse \( h \) can be calculated as follows:
\[
h = (42 - \sqrt{2}) \sqrt{2}
\]
Now, let's compute this:
\[
h = 42\sqrt{2} - 2
\]
Since this does not align exactly with any of the provided response options, let's evaluate the options:
1. The hypotenuse of the second triangle has length \( 42 - \sqrt{2} \).
2. The hypotenuse of the second triangle has length \( 4 \sqrt{2} \).
3. The hypotenuse of the second triangle has length \( 4 \).
4. The hypotenuse of the second triangle has length \( 8 \).
5. The hypotenuse of the second triangle has length \( 82 - \sqrt{2} \).
None of these answer choices match the calculated hypotenuse of \( 42\sqrt{2} - 2 \).
To simplify:
1. Calculate \( 42\sqrt{2} \):
- Approximate \( \sqrt{2} \) as \( 1.414 \):
- \( 42 \times 1.414 \approx 59.828 \).
- Therefore, \( 42\sqrt{2} - 2 \approx 57.828 \).
We can compare the calculated length to the provided options further but there's clearly no exact match.
**However**, from the computations and knowing that the original relationships are correctly applied, the answer options need to potentially identify what the response should reflect back to—hypotenuse in terms of \( x \) or further normalized terms. If specified in that structure, re-evaluating in exact calculative clarity can identify potential misunderstandings in normalization between computational observance and structured response.
Thus, with \( h = (42 - \sqrt{2}) \sqrt{2} \) leading to significant approximate identifiers, I would conclude:
**None. It’s not matching as given in the selections but \( 42\sqrt{2} - 2 \) remains isolative to clarity.**
But focusing on what we computed could indicate \( h \approx 57.828 \) as formed from \( 42\sqrt{2} - 2 \).
If direct assessment required relative proximity, it could loop back for engaging \( 8 \) via corresponding indicatives or \( 4\sqrt{2} \) as functional reference maybe linking a ratio.
**Final assertion would re-direct to hypotenuseriden once clarified \( 4\sqrt{2} < \approx h < 8 \). Please absolutely check and cross on terms relative practicality via hypotenuse relations.**
\[
h = x \sqrt{2}
\]
Now, apply this to the second triangle, where the side opposite the 45° angle is \( 42 - \sqrt{2} \).
Thus, the hypotenuse \( h \) can be calculated as follows:
\[
h = (42 - \sqrt{2}) \sqrt{2}
\]
Now, let's compute this:
\[
h = 42\sqrt{2} - 2
\]
Since this does not align exactly with any of the provided response options, let's evaluate the options:
1. The hypotenuse of the second triangle has length \( 42 - \sqrt{2} \).
2. The hypotenuse of the second triangle has length \( 4 \sqrt{2} \).
3. The hypotenuse of the second triangle has length \( 4 \).
4. The hypotenuse of the second triangle has length \( 8 \).
5. The hypotenuse of the second triangle has length \( 82 - \sqrt{2} \).
None of these answer choices match the calculated hypotenuse of \( 42\sqrt{2} - 2 \).
To simplify:
1. Calculate \( 42\sqrt{2} \):
- Approximate \( \sqrt{2} \) as \( 1.414 \):
- \( 42 \times 1.414 \approx 59.828 \).
- Therefore, \( 42\sqrt{2} - 2 \approx 57.828 \).
We can compare the calculated length to the provided options further but there's clearly no exact match.
**However**, from the computations and knowing that the original relationships are correctly applied, the answer options need to potentially identify what the response should reflect back to—hypotenuse in terms of \( x \) or further normalized terms. If specified in that structure, re-evaluating in exact calculative clarity can identify potential misunderstandings in normalization between computational observance and structured response.
Thus, with \( h = (42 - \sqrt{2}) \sqrt{2} \) leading to significant approximate identifiers, I would conclude:
**None. It’s not matching as given in the selections but \( 42\sqrt{2} - 2 \) remains isolative to clarity.**
But focusing on what we computed could indicate \( h \approx 57.828 \) as formed from \( 42\sqrt{2} - 2 \).
If direct assessment required relative proximity, it could loop back for engaging \( 8 \) via corresponding indicatives or \( 4\sqrt{2} \) as functional reference maybe linking a ratio.
**Final assertion would re-direct to hypotenuseriden once clarified \( 4\sqrt{2} < \approx h < 8 \). Please absolutely check and cross on terms relative practicality via hypotenuse relations.**