Question

1. points r, s, and t are located at r(-4,-5), s(2,-2), and t(8,1). (a) plot and label the three points. (b) draw $overline{rt}$. (c) are the points r, s, and t collinear? explain. (d) point u is located at u(-5,3). plot and label u. (e) draw $overline{su}$. (f) what type of angle - pair are $angle rsu$ and $angle tsu$? explain. (g) if $mangle rsu = 2x + 12$ and $mangle tsu = 5x - 7$, then algebraically solve for the value of x.

Explanation:

Step1: Plot points

On a coordinate - plane, for point $R(-4,-5)$, move 4 units to the left of the origin and 5 units down. For $S(2,-2)$, move 2 units to the right of the origin and 2 units down. For $T(8,1)$, move 8 units to the right of the origin and 1 unit up. Label them as $R$, $S$, and $T$.

Step2: Draw line - segment

Draw a straight line - segment connecting points $R$ and $T$ and label it as $\overline{RT}$.

Step3: Check collinearity

Find the slope between two pairs of points. The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
The slope of $\overline{RS}$ with $R(-4,-5)$ and $S(2,-2)$ is $m_{RS}=\frac{-2-(-5)}{2 - (-4)}=\frac{-2 + 5}{2 + 4}=\frac{3}{6}=\frac{1}{2}$.
The slope of $\overline{ST}$ with $S(2,-2)$ and $T(8,1)$ is $m_{ST}=\frac{1-(-2)}{8 - 2}=\frac{1 + 2}{8 - 2}=\frac{3}{6}=\frac{1}{2}$.
Since $m_{RS}=m_{ST}=\frac{1}{2}$ and point $S$ is common, points $R$, $S$, and $T$ are collinear.

Step4: Plot point U

For point $U(-5,3)$, move 5 units to the left of the origin and 3 units up and label it as $U$.

Step5: Draw line - segment

Draw a straight line - segment connecting points $S$ and $U$ and label it as $\overline{SU}$.

Step6: Identify angle pair

$\angle RSU$ and $\angle TSU$ are adjacent angles because they have a common side $\overrightarrow{SU}$ and a common vertex $S$, and they do not overlap.

Step7: Solve for x

Since $\angle RSU$ and $\angle TSU$ are adjacent angles and $\angle RST$ is a straight - line angle (assuming $\angle RST = 180^{\circ}$ as $R$, $S$, $T$ are collinear), then $m\angle RSU+m\angle TSU = 180^{\circ}$ (if considering the non - overlapping parts of the angles formed by the collinear points and the new point $U$). But if we assume that $\angle RST$ is not relevant and we just consider the relationship between $\angle RSU$ and $\angle TSU$ based on the given angle measures, and since they are adjacent, we have $m\angle RSU+m\angle TSU=m\angle RTU$. Also, if we assume that $\angle RSU$ and $\angle TSU$ are supplementary (a common case when dealing with adjacent angles formed in a geometric setup), we set up the equation:
$(2x + 12)+(5x-7)=180$.
Combine like terms: $2x+5x+12 - 7=180$, which simplifies to $7x + 5=180$.
Subtract 5 from both sides: $7x=180 - 5=175$.
Divide both sides by 7: $x=\frac{175}{7}=25$.

Answer:

(a) Points plotted as described.
(b) $\overline{RT}$ drawn.
(c) Yes, because the slopes of $\overline{RS}$ and $\overline{ST}$ are equal and they have a common point $S$.
(d) Point $U$ plotted as described.
(e) $\overline{SU}$ drawn.
(f) Adjacent angles because they have a common side $\overrightarrow{SU}$ and a common vertex $S$.
(g) $x = 25$