Question

I want to find step by step answers for these questions pls

Answer 1

Answers:

a) midpoint: (1, 4)

b) slope of PQ: 1/3

c) length of PQ: √40

d) equation of perpendicular bisector of PQ:

Explanation:

The given points are P(4, 5) and Q(-2, 3)

Part a)

The midpoint of two points (x1, y1) and (x2, y2) can be calculated as

$(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

So, replacing (x1, y1) = P(4, 5) and (x2, y2) = Q(-2, 3), we get:

$(\frac{4+(-2)}{2},\frac{5+3}{2})=(\frac{2}{2},\frac{8}{2})=(1,4)$

Then, the midpoint of PQ is (1, 4)

Part b)

The slope of a segment that passes through points (x1, y1) and (x2, y2) can be calculated as

$\text{slope}=\frac{y_2-y_1}{x_2-x_1}$

Replacing (x1, y1) = P(4, 5) and (x2, y2) = Q(-2, 3), we get:

$\text{slope}=\frac{3-5}{-2-4}=\frac{-2}{-6}=\frac{1}{3}$

So, the slope is 1/3

Part c)

The length of a segment that goes from (x1, y1) to (x2, y2) is calculated as

$\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}$

Replacing (x1, y1) = P(4, 5) and (x2, y2) = Q(-2, 3), we get:

$\begin{gathered} \sqrt[]{(-2-4)^2+(3-5)^2} \\ \sqrt[]{(-6_{})^2+(-2)^2} \\ \sqrt[]{36+4} \\ \sqrt[]{40} \end{gathered}$

Therefore, the length of PQ is √40.

Part d)

The perpendicular bisector of PQ is a line that divides the segment into two equal and forms a 90 degrees angle with the segment.

First, we need to calculate the slope of the line. Taking into account that the slope of perpendicular lines multiply to -1, we get that the slope of the perpendicular bisector is

$\begin{gathered} m\cdot\frac{1}{3}=-1 \\ m\cdot\frac{1}{3}\cdot3=-1\cdot3 \\ m=-3 \end{gathered}$

Additionally, the perpendicular bisector will pass through the midpoint of PQ, so it will pass through (1, 4).

Now, the equation of a line with slope m= -3 that passes through the point (x1, y1) = (1, 4) is

$\begin{gathered} y-y_1=m(x-x_1) \\ y-4=-3(x-1) \\ y-4=-3x-3(-1) \\ y-4=-3x+3 \\ y-4+4--3x_{}+3+4 \\ y=-3x+7 \end{gathered}$

Therefore, the equation of the perpendicular bisector is y = -3x + 7.