Question

given: line f includes points (1,9) and (3,1). line h includes points (-2,3) and (10,6). prove: line f ⊥ line h. which is a key statement in the proof? the slope of line h is - 1/4. the slope of line f is - 4. the distance between (-2,3) and (10,6) is √73. the distance between (1,9) and (3,1) is √68.

Explanation:

Step1: Recall slope - formula

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Calculate slope of line $h$

For line $h$ with points $(-2,3)$ and $(10,6)$, $m_h=\frac{6 - 3}{10-(-2)}=\frac{3}{12}=\frac{1}{4}$.

Step3: Calculate slope of line $f$

For line $f$ with points $(1,9)$ and $(3,1)$, $m_f=\frac{1 - 9}{3 - 1}=\frac{-8}{2}=-4$.

Step4: Recall perpendicular - line property

Two non - vertical lines are perpendicular if and only if the product of their slopes is $- 1$. Here, $m_f\times m_h=-4\times\frac{1}{4}=-1$.

Answer:

The key statement in the proof is that the slope of line $h$ is $\frac{1}{4}$ and the slope of line $f$ is $-4$ (since the product of their slopes being $-1$ is used to prove that line $f\perp$ line $h$).