To verify that triangle [tex]\(WXY\)[/tex] is a right triangle using the concept of slopes, we need to determine which pair of lines form a right angle. A right angle is formed when the product of the slopes of two intersecting lines is [tex]\(-1\)[/tex]. These two slopes are known as opposite reciprocals.
Here are the slopes given:
- 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].
Next, we'll check each statement to see which one verifies that the triangle [tex]\(WXY\)[/tex] is a right triangle:
1. The slopes of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{YW}\)[/tex] are opposite reciprocals:
- Multiplying the slopes of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{YW}\)[/tex]: [tex]\(-\frac{2}{5} \times \frac{5}{2} = -1\)[/tex].
- Since the product is [tex]\(-1\)[/tex], these two lines are perpendicular to each other.
2. The slopes of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{XY}\)[/tex] are opposite reciprocals:
- Multiplying the slopes of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{XY}\)[/tex]: [tex]\(-\frac{2}{5} \times 0.56 \neq -1\)[/tex].
- Since the product is not [tex]\(-1\)[/tex], these two lines are not perpendicular.
3. The slopes of [tex]\(\overline{XY}\)[/tex] and [tex]\(\overline{WX}\)[/tex] have opposite signs:
- While it is true that [tex]\(\overline{XY}\)[/tex] (0.56) and [tex]\(\overline{WX}\)[/tex] (-\frac{2}{5}) have opposite signs, this does not verify that they are perpendicular.
4. The slopes of [tex]\(\overline{XY}\)[/tex] and [tex]\(\overline{YW}\)[/tex] have the same signs:
- Both slopes ([tex]\(0.56\)[/tex] and [tex]\(\frac{5}{2}\)[/tex]) are positive, but this does not indicate that these two lines form a right angle.
From the above analysis, we can determine that the correct statement is:
The slopes of [tex]\(\overline{WX}\)[/tex] and [tex]\(\overline{YW}\)[/tex] are opposite reciprocals.
Thus, this verifies that triangle [tex]\(WXY\)[/tex] is a right triangle with a right angle at point [tex]\(X\)[/tex].
