Question

applying theorems to determine slope. if line b is perpendicular to line a, and line c is perpendicular to line a, what is the slope of line c? (m_c=) 1 (a blank box)

Explanation:

Step1: Recall perpendicular - line slope relation

If two non - vertical lines are perpendicular, the product of their slopes is - 1. Let the slope of line $a$ be $m_a$, the slope of line $b$ be $m_b$, and the slope of line $c$ be $m_c$. We know that $m_a\times m_b=-1$ and $m_a\times m_c = - 1$.

Step2: Deduce the relation between $m_b$ and $m_c$

From $m_a\times m_b=-1$ and $m_a\times m_c=-1$, we can conclude that $m_b = m_c$ (since $m_a
eq0$ for non - vertical lines. If $m_a = 0$, the lines $b$ and $c$ would be vertical and slope would be undefined, but from the graph they are not vertical).

Step3: Find the slope of line $b$ (or $c$) using two - point formula

Let's assume two points on line $b$: say $(0,0)$ and $(1,1)$. The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For the points $(x_1,y_1)=(0,0)$ and $(x_2,y_2)=(1,1)$, we have $m=\frac{1 - 0}{1 - 0}=1$. Since $m_b = m_c$, the slope of line $c$ is also 1.

Answer:

1