Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Discover a wealth of knowledge from experts across different disciplines on our comprehensive Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To verify that triangle [tex]\(WXY\)[/tex] is a right triangle, we need to investigate the relationships between the slopes of its sides. The slopes given are:
- The slope of [tex]\(\overline{WX}\)[/tex] is [tex]\(-\frac{2}{5}\)[/tex].
- The slope of [tex]\(\overline{XY}\)[/tex] is [tex]\(0.56\)[/tex].
- The slope of [tex]\(\overline{YW}\)[/tex] is [tex]\(\frac{5}{2}\)[/tex].
### Checking Each Statement
1. The slopes of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{YW}\)[/tex] are opposite reciprocals.
Verification:
[tex]\[ -\frac{2}{5} \cdot \frac{5}{2} = -1 \][/tex]
The product of the slopes is [tex]\(-1\)[/tex], confirming that [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{YW}\)[/tex] are perpendicular because they are opposite reciprocals. This suggests that [tex]\(\angle WXY\)[/tex] is a right angle.
Result: True
2. The slopes of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{XY}\)[/tex] are opposite reciprocals.
Verification:
Since the slopes are [tex]\(-\frac{2}{5}\)[/tex] and [tex]\(0.56\)[/tex]:
[tex]\[ -\frac{2}{5} \cdot \left(-\frac{1}{0.56}\right) = -1 \quad\text{to check opposite reciprocals}. \][/tex]
However, simplifying:
[tex]\[ -\frac{2}{5} \cdot -1.7857 \approx 0.714 \neq -1 \][/tex]
Thus, they are not opposite reciprocals.
Result: False
3. The slopes of [tex]\(\overline{XY}\)[/tex] and [tex]\(\overline{WX}\)[/tex] have opposite signs.
Verification:
[tex]\[ \text{Slope of } \overline{XY} = 0.56 \quad (\text{positive}) \][/tex]
[tex]\[ \text{Slope of } \overline{WX} = -\frac{2}{5} \quad (\text{negative}) \][/tex]
Since one is positive and the other is negative, they indeed have opposite signs.
Result: True
4. The slopes of [tex]\(\overline{XY}\)[/tex] and [tex]\(\overline{YW}\)[/tex] have the same signs.
Verification:
[tex]\[ \text{Slope of } \overline{XY} = 0.56 \quad (\text{positive}) \][/tex]
[tex]\[ \text{Slope of } \overline{YW} = \frac{5}{2} \quad (\text{positive}) \][/tex]
Both slopes are positive, indicating they have the same signs.
Result: True
### Conclusion
The statement that verifies that triangle [tex]\(WXY\)[/tex] is a right triangle is:
The slopes of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{YW}\)[/tex] are opposite reciprocals.
This relationship confirms that the angle between [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{YW}\)[/tex] is [tex]\(90\)[/tex] degrees, which is the defining property of a right triangle.
- The slope of [tex]\(\overline{WX}\)[/tex] is [tex]\(-\frac{2}{5}\)[/tex].
- The slope of [tex]\(\overline{XY}\)[/tex] is [tex]\(0.56\)[/tex].
- The slope of [tex]\(\overline{YW}\)[/tex] is [tex]\(\frac{5}{2}\)[/tex].
### Checking Each Statement
1. The slopes of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{YW}\)[/tex] are opposite reciprocals.
Verification:
[tex]\[ -\frac{2}{5} \cdot \frac{5}{2} = -1 \][/tex]
The product of the slopes is [tex]\(-1\)[/tex], confirming that [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{YW}\)[/tex] are perpendicular because they are opposite reciprocals. This suggests that [tex]\(\angle WXY\)[/tex] is a right angle.
Result: True
2. The slopes of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{XY}\)[/tex] are opposite reciprocals.
Verification:
Since the slopes are [tex]\(-\frac{2}{5}\)[/tex] and [tex]\(0.56\)[/tex]:
[tex]\[ -\frac{2}{5} \cdot \left(-\frac{1}{0.56}\right) = -1 \quad\text{to check opposite reciprocals}. \][/tex]
However, simplifying:
[tex]\[ -\frac{2}{5} \cdot -1.7857 \approx 0.714 \neq -1 \][/tex]
Thus, they are not opposite reciprocals.
Result: False
3. The slopes of [tex]\(\overline{XY}\)[/tex] and [tex]\(\overline{WX}\)[/tex] have opposite signs.
Verification:
[tex]\[ \text{Slope of } \overline{XY} = 0.56 \quad (\text{positive}) \][/tex]
[tex]\[ \text{Slope of } \overline{WX} = -\frac{2}{5} \quad (\text{negative}) \][/tex]
Since one is positive and the other is negative, they indeed have opposite signs.
Result: True
4. The slopes of [tex]\(\overline{XY}\)[/tex] and [tex]\(\overline{YW}\)[/tex] have the same signs.
Verification:
[tex]\[ \text{Slope of } \overline{XY} = 0.56 \quad (\text{positive}) \][/tex]
[tex]\[ \text{Slope of } \overline{YW} = \frac{5}{2} \quad (\text{positive}) \][/tex]
Both slopes are positive, indicating they have the same signs.
Result: True
### Conclusion
The statement that verifies that triangle [tex]\(WXY\)[/tex] is a right triangle is:
The slopes of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{YW}\)[/tex] are opposite reciprocals.
This relationship confirms that the angle between [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{YW}\)[/tex] is [tex]\(90\)[/tex] degrees, which is the defining property of a right triangle.
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.