To determine if two lines are perpendicular, we need to check the product of their slopes. The general condition for two lines to be perpendicular is that the product of their slopes equals [tex]\(-1\)[/tex].
Given:
- The slope of Line [tex]\( A \)[/tex] is [tex]\(\frac{2}{7}\)[/tex].
- The slope of Line [tex]\( B \)[/tex] is [tex]\(-\frac{7}{2}\)[/tex].
Step-by-Step Solution:
1. Multiply the slopes of the given lines:
[tex]\[
\frac{2}{7} \times -\frac{7}{2}
\][/tex]
2. When multiplying these fractions, multiply the numerators together and the denominators together:
[tex]\[
\frac{2 \times -7}{7 \times 2}
\][/tex]
3. This simplifies as follows:
[tex]\[
\frac{-14}{14}
\][/tex]
4. Simplify the fraction:
[tex]\[
\frac{-14}{14} = -1
\][/tex]
Since the product of the slopes is [tex]\(-1\)[/tex], this confirms that the lines are perpendicular.
Therefore, the answer is:
- The product of the slopes is [tex]\(-1\)[/tex].
- The lines are perpendicular: True.
