Question

1.09: swbat write the equation for and graph a perpendicular bisector on the coordinate plane. directions: complete the following questions by showing all work and annotations. keep work organized and box any final answer. all work must be shown in order to receive full credit. #1.) mn has vertices m(-7,1), n(1,-5). line d is the perpendicular bisector of segment mn. a.) graph segment mn on the coordinate plane. b.) what is the equation of line d? c.) a. (y - 2)=\frac{4}{3}(x + 3) b. (y + 2)=\frac{4}{3}(x + 3) c. (y - 2)=-\frac{3}{4}(x + 3) d. (y + 2)=-\frac{3}{4}(x + 3) c.) *write the equation in slope intercept form. d.) graph line d to confirm it is the perpendicular bisector.

Explanation:

Step1: Find the mid - point of segment MN

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. For $M(-7,1)$ and $N(1,-5)$, we have $x=\frac{-7 + 1}{2}=\frac{-6}{2}=-3$ and $y=\frac{1+( - 5)}{2}=\frac{1 - 5}{2}=-2$. So the mid - point is $(-3,-2)$.

Step2: Find the slope of segment MN

The slope formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For $M(-7,1)$ and $N(1,-5)$, $m_{MN}=\frac{-5 - 1}{1-( - 7)}=\frac{-6}{8}=-\frac{3}{4}$.

Step3: Find the slope of the perpendicular bisector

If two lines are perpendicular, the product of their slopes is $- 1$. Let the slope of the perpendicular bisector be $m_d$. Then $m_d\times m_{MN}=-1$. Since $m_{MN}=-\frac{3}{4}$, we have $m_d=\frac{4}{3}$.

Step4: Write the point - slope form of the equation of line d

The point - slope form of a line is $y - y_0=m(x - x_0)$, where $(x_0,y_0)$ is a point on the line and $m$ is the slope. Using the mid - point $(-3,-2)$ and slope $m_d=\frac{4}{3}$, the equation is $y+2=\frac{4}{3}(x + 3)$.

Step5: Write the slope - intercept form of the equation of line d

Starting with $y+2=\frac{4}{3}(x + 3)$, we distribute the $\frac{4}{3}$: $y+2=\frac{4}{3}x+4$. Then subtract 2 from both sides to get $y=\frac{4}{3}x + 2$.

Answer:

b. $(y + 2)=\frac{4}{3}(x + 3)$
$y=\frac{4}{3}x+2$

(For graphing parts a and d, for part a, plot the points $M(-7,1)$ and $N(1,-5)$ and draw the line segment between them. For part d, plot the line $y=\frac{4}{3}x + 2$ which has a y - intercept of 2 and a slope of $\frac{4}{3}$ and visually confirm it bisects and is perpendicular to segment MN)