Question

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

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 $m$

Using points $(-2,7)$ and $(10,5)$, $m_m=\frac{5 - 7}{10-(-2)}=\frac{-2}{12}=-\frac{1}{6}$.

Step3: Calculate slope of line $n$

Using points $(-5,0)$ and $(0,-2)$, $m_n=\frac{-2 - 0}{0-(-5)}=-\frac{2}{5}$.

Step4: Calculate slope of line $p$

Using points $(3,5)$ and $(6,15)$, $m_p=\frac{15 - 5}{6 - 3}=\frac{10}{3}$.

Step5: Calculate slope of line $q$

Using points $(3,5)$ and $(10,5)$, $m_q=\frac{5 - 5}{10 - 3}=0$.

Step6: Recall perpendicular and parallel line conditions

Two lines are parallel if their slopes are equal ($m_1=m_2$), and two lines are perpendicular if the product of their slopes is $- 1$ ($m_1\times m_2=-1$).

Step7: Check option A

$m_p\times m_q=\frac{10}{3}\times0 = 0
eq - 1$, so $p$ is not perpendicular to $q$.

Step8: Check option B

$m_q\times m_n=0\times(-\frac{2}{5}) = 0
eq - 1$, so $q$ is not perpendicular to $n$.

Step9: Check option C

$m_m=-\frac{1}{6}$ and $m_n=-\frac{2}{5}$, $m_m
eq m_n$, so $m$ is not parallel to $n$.

Step10: Check option D

$m_p\times m_m=\frac{10}{3}\times(-\frac{1}{6})=-\frac{5}{9}
eq - 1$, so $p$ is not perpendicular to $m$.

Step11: Check option E

$m_m=-\frac{1}{6}$ and $m_q = 0$, $m_m
eq m_q$, so $m$ is not parallel to $q$.

Step12: Check option F

$m_n=-\frac{2}{5}$ and $m_q = 0$, $m_n
eq m_q$, so $n$ is not parallel to $q$.

Answer:

None of the statements A - F are true.