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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.
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