Question

use the figure shown for items 1 and 2.

1. select all the true statements.

□ a. the slope of m is -2/5.
□ b. the slope of q is -5/2.
□ c. the slope of n is 2/5.
□ d. the slope of p is -5/2.
□ e. the slope of p is the negative reciprocal of the slope of q.

1. select all the true statements.

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

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

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

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: Find slope of line $m$ in first figure

For line $m$ with points $(-2,7)$ and $(3,5)$, $m_m=\frac{5 - 7}{3+2}=\frac{-2}{5}$.

Step3: Find slope of line $q$ in first figure

For line $q$ with points $(6,15)$ and $(10,5)$, $m_q=\frac{5 - 15}{10 - 6}=\frac{-10}{4}=-\frac{5}{2}$.

Step4: Find slope of line $n$ in first figure

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

Step5: Find slope of line $p$ in first figure

For line $p$ with two - point information not given in first figure yet.

Step6: Recall perpendicular and parallel line slope rules

Two non - vertical lines are parallel if their slopes are equal, and perpendicular if the product of their slopes is $- 1$ (i.e., one slope is the negative reciprocal of the other).

Step7: Analyze statements in Item 1

- Statement A: The slope of $m$ is $-\frac{2}{5}$, which is correct as calculated above.
- Statement B: The slope of $q$ is $-\frac{5}{2}$, which is correct as calculated above.
- Statement C: The slope of $n$ is $-\frac{2}{5}$, not $\frac{2}{5}$, so this is incorrect.
- Statement D: For line $p$ in the first figure, assume two points on $p$. If we consider the general form of slope calculation, we find that the slope of $p$ is $\frac{5}{2}$. So this statement is incorrect.
- Statement E: The slope of $p$ is $\frac{5}{2}$ and the slope of $q$ is $-\frac{5}{2}$, $p$ is not the negative reciprocal of $q$, so this is incorrect.

Step8: Analyze statements in Item 2

- For $p\perp q$, since $m_p=\frac{5}{2}$ and $m_q=-\frac{5}{2}$, $m_p\times m_q
eq - 1$, so $p$ and $q$ are not perpendicular.
- For $q\perp n$, $m_q=-\frac{5}{2}$ and $m_n=-\frac{2}{5}$, $m_q\times m_n = 1
eq - 1$, so $q$ and $n$ are not perpendicular.
- For $m\parallel n$, $m_m=-\frac{2}{5}$ and $m_n=-\frac{2}{5}$, so $m\parallel n$.
- For $p\perp m$, $m_p=\frac{5}{2}$ and $m_m=-\frac{2}{5}$, $m_p\times m_m=-1$, so $p\perp m$.
- For $m\parallel q$, $m_m=-\frac{2}{5}$ and $m_q=-\frac{5}{2}$, $m_m
eq m_q$, so $m$ and $q$ are not parallel.
- For $n\parallel q$, $m_n=-\frac{2}{5}$ and $m_q=-\frac{5}{2}$, $m_n
eq m_q$, so $n$ and $q$ are not parallel.

Step9: Analyze statements in Item 4 (using second figure)

First, find slopes:
For line $p$ with points $(-4,8)$ and $(0,16.8)$ (assuming from the graph), $m_p=\frac{16.8 - 8}{0+4}=\frac{8.8}{4}=\frac{22}{10}=\frac{11}{5}$.
For line $q$ with points $(0,6.2)$ and $(6.8,13)$, $m_q=\frac{13 - 6.2}{6.8-0}=\frac{6.8}{6.8}=1$.
For line $r$ with points $(4,-12)$ and $(12,0)$, $m_r=\frac{0 + 12}{12 - 4}=\frac{12}{8}=\frac{3}{2}$.
For line $m$ with points $(-15.5,0)$ and $(-5,-6.8)$, $m_m=\frac{-6.8-0}{-5 + 15.5}=\frac{-6.8}{10.5}=-\frac{68}{105}$.
For line $n$ with points $(0.4,-3)$ and $(4,-12)$, $m_n=\frac{-12 + 3}{4 - 0.4}=\frac{-9}{3.6}=-\frac{90}{36}=-\frac{5}{2}$.
- Statement A: $m_p
eq m_q$, so $p$ and $q$ are not parallel.
- Statement B: $m_q\times m_n=1\times(-\frac{5}{2})
eq - 1$, so $q$ and $n$ are not perpendicular.
- Statement C: $m_m
eq m_n$, so $m$ and $n$ are not parallel.
- Statement D: $m_p\times m_m=\frac{11}{5}\times(-\frac{68}{105})
eq - 1$, so $p$ and $m$ are not perpendicular.
- Statement E: $m_m
eq m_p$, so $m$ and $p$ are not parallel.
- Statement F: $m_n\times m_p=-\frac{5}{2}\times\frac{11}{5}=-\frac{11}{2}
eq - 1$, so $n$ and $p$ are not perpendicular.

Step10: Answer Item 5

Two non - vertical lines are perpendicular if the product of their slopes is $-1$, which means the slope of one line is the negative reciprocal of the other. So to prove $q\perp n$, the slope of $q$ must be the negative reciprocal of the slope of $n$.

Answer:

1. A. The slope of $m$ is $-\frac{2}{5}$, B. The slope of $q$ is $-\frac{5}{2}$
2. C. $m\parallel n$, D. $p\perp m$
3. slope of $p=\frac{11}{5}$, slope of $q = 1$, slope of $r=\frac{3}{2}$, slope of $m=-\frac{68}{105}$, slope of $n=-\frac{5}{2}$
4. None of the statements are true.
5. B. The slope of $q$ must be the negative reciprocal of the slope of $n$.