Question

1. if $overline{ab}$ is defined by the endpoints a(4,2) and b(8,6), write an equation of the line that is the perpendicular bisector of $overline{ab}$.
2. the endpoints of $overline{ab}$ and $overline{cd}$ are a(-3,6), b(5,4), c(-7,3) and d(9,7). determine if $overline{ab}$ and $overline{cd}$ bisect each other.
3. point p divides the directed line segment from point a(-4,-1) to point b(6,4) in the ratio 2:3. find the coordinates of point p.

Explanation:

Step1: Find slope of line segment AB

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $A(4,2)$ and $B(8,6)$, we have $m_{AB}=\frac{6 - 2}{8 - 4}=\frac{4}{4}=1$.

Step2: Find perpendicular slope

The perpendicular slope $m_{\perp}$ is the negative - reciprocal of the slope of AB. So $m_{\perp}=- 1$.

Step3: Find mid - point of AB

The mid - point formula is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. For $A(4,2)$ and $B(8,6)$, the mid - point $M=(\frac{4 + 8}{2},\frac{2+6}{2})=(6,4)$.

Step4: Write equation of the perpendicular bisector

Using the point - slope form of a line $y - y_0=m(x - x_0)$, where $(x_0,y_0)$ is the mid - point $(6,4)$ and $m=-1$. We get $y - 4=-1(x - 6)$, which simplifies to $y=-x + 10$.

for problem 4:

Step1: Find mid - point of AB

For $A(-3,6)$ and $B(5,4)$, using the mid - point formula $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$, we have $M_{AB}=(\frac{-3 + 5}{2},\frac{6+4}{2})=(1,5)$.

Step2: Find mid - point of CD

For $C(-7,3)$ and $D(9,7)$, using the mid - point formula, we have $M_{CD}=(\frac{-7 + 9}{2},\frac{3 + 7}{2})=(1,5)$.

Step3: Conclusion

Since the mid - points of AB and CD are the same $(1,5)$, the line segments AB and CD bisect each other.

for problem 5:

Step1: Use the section formula

The section formula for a point $P(x,y)$ that divides the line segment from $A(x_1,y_1)$ to $B(x_2,y_2)$ in the ratio $m:n$ is $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here $A(-4,-1)$, $B(6,4)$, and the ratio $m:n = 2:3$.
$x=\frac{2\times6+3\times(-4)}{2 + 3}=\frac{12-12}{5}=0$.
$y=\frac{2\times4+3\times(-1)}{2 + 3}=\frac{8 - 3}{5}=1$.

Answer:

$y=-x + 10$