Question

select all the true statements.
a. p||q
b. q⊥n
c. m||n
d. p⊥m

Explanation:

Step1: Recall slope - formula

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line.

Step2: Calculate slope of line $p$

Using points $(-4,8)$ and $(0.4,-3)$ on line $p$, $m_p=\frac{-3 - 8}{0.4+4}=\frac{-11}{4.4}=- 2.5$.

Step3: Calculate slope of line $q$

Using points $(0,6.2)$ and $(6.8,13)$ on line $q$, $m_q=\frac{13 - 6.2}{6.8-0}=\frac{6.8}{6.8}=1$.

Step4: Calculate slope of line $n$

Using points $(-5,-6.8)$ and $(4,-12)$ on line $n$, $m_n=\frac{-12 + 6.8}{4 + 5}=\frac{-5.2}{9}\approx - 0.58$.

Step5: Calculate slope of line $m$

Using points $(-15.5,0)$ and $(12,0)$ on line $m$, $m_m=\frac{0-0}{12 + 15.5}=0$.

Step6: Check parallel and perpendicular conditions

• Two lines are parallel if their slopes are equal.
• Two lines are perpendicular if the product of their slopes is $- 1$.
• For option A: $m_p=-2.5$ and $m_q = 1$, $m_p

eq m_q$, so $p$ is not parallel to $q$.

• For option B: $m_q\times m_n=1\times(-0.58)

eq - 1$, so $q$ is not perpendicular to $n$.

• For option C: $m_m = 0$ and $m_n\approx - 0.58$, $m_m

eq m_n$, so $m$ is not parallel to $n$.

• For option D: $m_p\times m_m=-2.5\times0 = 0$. Lines with slope $m_p=-2.5$ and $m_m = 0$ (horizontal - vertical relationship in a sense) are perpendicular.

Answer:

D. $p\perp m$