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 $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Calculate slope of line $p$

For line $p$ with points $(3,5)$ and $(6,15)$, $m_p=\frac{15 - 5}{6 - 3}=\frac{10}{3}$.

Step3: Calculate slope of line $q$

For line $q$ with points $(0,- 2)$ and $(10,5)$, $m_q=\frac{5+2}{10 - 0}=\frac{7}{10}$.

Step4: Calculate slope of line $m$

For line $m$ with points $(-2,7)$ and $(10,5)$, $m_m=\frac{5 - 7}{10 + 2}=-\frac{1}{6}$.

Step5: Calculate slope of line $n$

For line $n$ with points $(-5,0)$ and $(0,-2)$, $m_n=\frac{-2-0}{0 + 5}=-\frac{2}{5}$.

Step6: Check perpendicular and parallel conditions

Two lines are perpendicular if the product of their slopes is $- 1$, and parallel if their slopes are equal.

• $m_p\times m_q=\frac{10}{3}\times\frac{7}{10}=\frac{7}{3}

eq - 1$, so $p$ and $q$ are not perpendicular.

• $m_q\times m_n=\frac{7}{10}\times(-\frac{2}{5})=-\frac{7}{25}

eq - 1$, so $q$ and $n$ are not perpendicular.

• $m_m=m_n=-\frac{1}{6}$, so $m\parallel n$.
• $m_p\times m_m=\frac{10}{3}\times(-\frac{1}{6})=-\frac{5}{9}

eq - 1$, so $p$ and $m$ are not perpendicular.

• $m_m

eq m_q$, so $m$ and $q$ are not parallel.

• $m_n

eq m_q$, so $n$ and $q$ are not parallel.

Answer:

C. $m\parallel n$