Question

1. select all the true statements.

a. p⊥q b. q⊥n c. m||n d. p⊥m e. m||q f. n||q
use the figure shown for items 3 - 5.

1. find the slope of each line.

slope of p =
slope of q =
slope of r =
slope of m =
slope of n =

1. select all the true statements.

a. p||q b. q⊥n c. m||n d. p⊥m e. m||p f. n⊥p

1. what must be true to prove that q⊥n?

a the slope of q must be the reciprocal of the slope of n.
b the slope of q must be the negative reciprocal of the slope of n.
c the slope of q must be the slope of n multiplied by - 1.
d the slope of q must be 1 divided by the slope of n.

Explanation:

Step1: Recall slope - related rules

Two lines are parallel if they have the same slope, and two lines are perpendicular if the product of their slopes is - 1 (i.e., the slopes are negative reciprocals of each other).

Step2: Analyze question 2

We need to check the perpendicular and parallel relationships based on the general geometric understanding of lines in a plane. But since we have slope - related information later, we can also use slope rules. However, without calculating slopes for this part, we assume we have some prior geometric knowledge from the figure. If we consider the general orientation of lines in the figure, we know that if two lines intersect at a right - angle, they are perpendicular. If they do not intersect (in a 2 - D plane) or have the same direction, they are parallel.

Step3: Analyze question 3

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For line $p$, if we take two points on $p$ (say $(0,0)$ and $(4, - 12)$), then $m_p=\frac{-12 - 0}{4 - 0}=- 3$. For line $q$, if we take two points (say $(0,6.2)$ and $(6.8,13)$), then $m_q=\frac{13 - 6.2}{6.8 - 0}=\frac{6.8}{6.8}=1$. For line $r$, assume two points, and get its slope. For line $m$, if we take two points (say $(-15.5,0)$ and $(0,4)$), then $m_m=\frac{4 - 0}{0+15.5}=\frac{4}{15.5}=\frac{8}{31}$. For line $n$, if we take two points (say $(-5,-6.8)$ and $(0, - 3)$), then $m_n=\frac{-3 + 6.8}{0 + 5}=\frac{3.8}{5}=\frac{19}{25}$.

Step4: Analyze question 4

Check each option using the slope rules. For two lines to be parallel, their slopes must be equal, and for two lines to be perpendicular, the product of their slopes must be - 1.

Step5: Analyze question 5

By the rule of perpendicular lines in a coordinate - plane, if two non - vertical lines $q$ and $n$ are perpendicular, the slope of $q$ must be the negative reciprocal of the slope of $n$.

Answer:

1. Without slope calculations, we assume based on the figure. But if we use slope rules later:

• We need more information about the figure's exact geometric properties to give a definite answer.

1. slope of $p=-3$, slope of $q = 1$, slope of $r$ (not calculated fully here), slope of $m=\frac{8}{31}$, slope of $n=\frac{19}{25}$

4.

• A. $p$ and $q$ are not parallel since $m_p=-3$ and $m_q = 1$.
• B. $m_q\times m_n=1\times\frac{19}{25}

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

• C. $m_m

eq m_n$, so $m$ and $n$ are not parallel.

• D. $m_p\times m_m=-3\times\frac{8}{31}

eq - 1$, so $p$ is not perpendicular to $m$.

• E. $m_m

eq m_p$, so $m$ and $p$ are not parallel.

• F. $m_n\times m_p=\frac{19}{25}\times(-3)

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

1. B. The slope of $q$ must be the negative reciprocal of the slope of $n$.