To determine if triangle WXY is a right triangle, we need to explore the relationship between the slopes of its sides. Specifically, in a right triangle, the slopes of the two perpendicular sides must be opposite reciprocals of each other. Let's examine each statement in the context of this information.
Given:
- The slope of [tex]\(\overline{WX}\)[/tex] is [tex]\(-\frac{2}{5}\)[/tex].
- The slope of [tex]\(\overline{XY}\)[/tex] is 0.56.
- The slope of [tex]\(\overline{YW}\)[/tex] is [tex]\(\frac{5}{2}\)[/tex].
Step-by-Step Analysis:
1. Checking if the slopes of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{YW}\)[/tex] are opposite reciprocals:
[tex]\[ \text{slope of } \overline{WX} = -\frac{2}{5} \][/tex]
[tex]\[ \text{slope of } \overline{YW} = \frac{5}{2} \][/tex]
Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]:
[tex]\[ \left(-\frac{2}{5}\right) \cdot \left(\frac{5}{2}\right) = -\frac{2 \times 5}{5 \times 2} = -\frac{10}{10} = -1 \][/tex]
Therefore, the slopes [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{YW}\)[/tex] are indeed opposite reciprocals.
2. Checking if the slopes of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{XY}\)[/tex] are opposite reciprocals:
[tex]\[ \text{slope of } \overline{WX} = -\frac{2}{5} \][/tex]
[tex]\[ \text{slope of } \overline{XY} = 0.56 \][/tex]
We convert 0.56 to a fraction for consistency in easy assessment:
[tex]\[ 0.56 = \frac{28}{50} = \frac{14}{25} \][/tex]
Now, we check if the product of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{XY}\)[/tex] equals [tex]\(-1\)[/tex]:
[tex]\[ \left(-\frac{2}{5}\right) \cdot \left(\frac{14}{25}\right) = -\frac{28}{125} \neq -1 \][/tex]
Thus, the slopes [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{XY}\)[/tex] are not opposite reciprocals.
3. Checking if the slopes of [tex]\(\overline{XY}\)[/tex] and [tex]\(\overline{WX}\)[/tex] have opposite signs:
The slope of [tex]\(\overline{XY}\)[/tex] is positive (0.56) and the slope of [tex]\(\overline{WX}\)[/tex] is negative ([tex]\(-\frac{2}{5}\)[/tex]). So they indeed have opposite signs. However, this check alone is insufficient to confirm that the triangle is a right triangle.
4. Checking if the slopes of [tex]\(\overline{XY}\)[/tex] and [tex]\(\overline{YW}\)[/tex] have the same signs:
[tex]\[ \text{slope of } \overline{XY} = 0.56 \,(\text{positive}) \][/tex]
[tex]\[ \text{slope of } \overline{YW} = \frac{5}{2} \,(\text{positive}) \][/tex]
Both slopes are positive and hence have the same signs. This, however, does not verify the presence of a right angle in the triangle.
Upon analyzing all statements, the first statement correctly identifies the condition necessary to verify that triangle WXY is a right triangle. The slopes of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{YW}\)[/tex] are opposite reciprocals, proving that these sides are perpendicular and thus forming a right angle.
Therefore, the correct statement is:
1. The slopes of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{YW}\)[/tex] are opposite reciprocals.
