Question

Write an equation of the perpendicular bisector of the segment with the given endpoints.

( E and F are two different questions )
E. A(-2,8) B(-4,-6)

F. X(5,17) Y(7,3)​

Answer 1

Explanation:

E. To find the equation of the perpendicular bisector of the segment with endpoints A(-2,8) and B(-4,-6), we need to follow these steps:

Find the midpoint of the segment AB using the midpoint formula:

Midpoint = [ (x1 + x2)/2 , (y1 + y2)/2 ]

Midpoint = [ (-2 - 4)/2 , (8 - 6)/2 ]

Midpoint = [ -3 , 1 ]

Therefore, the midpoint of AB is (-3, 1).

Find the slope of the line passing through the points A and B using the slope formula:

Slope = (y2 - y1) / (x2 - x1)

Slope = (-6 - 8) / (-4 - (-2))

Slope = (-14) / (-2)

Slope = 7

Since we need the slope of the perpendicular bisector, we can use the fact that the product of the slopes of two perpendicular lines is -1. Therefore, the slope of the perpendicular bisector is the negative reciprocal of the slope of AB:

Slope of perpendicular bisector = -1/7

Finally, we can use the point-slope form of a line to write the equation of the perpendicular bisector with slope -1/7 and passing through the midpoint (-3, 1):

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is a point on the line.

Substituting m = -1/7 and (x1, y1) = (-3, 1), we get:

y - 1 = (-1/7)(x - (-3))

Simplifying, we get:

y - 1 = (-1/7)x - 3/7

y = (-1/7)x - 3/7 + 1

y = (-1/7)x + 4/7

Therefore, the equation of the perpendicular bisector of the segment with endpoints A(-2,8) and B(-4,-6) is y = (-1/7)x + 4/7

F. To find the perpendicular bisector of segment XY, we first need to find the midpoint of the segment. Using the midpoint formula, we get:

Midpoint = ( (x1+x2)/2 , (y1+y2)/2 )

Midpoint = ( (5+7)/2 , (17+3)/2 )

Midpoint = ( 6 , 10 )

So the midpoint of segment XY is (6,10). Now we need to find the slope of segment XY. Using the slope formula, we get:

Slope of XY = (y2 - y1) / (x2 - x1)

Slope of XY = (3 - 17) / (7 - 5)

Slope of XY = -7

The perpendicular bisector of XY will have a slope of the negative reciprocal of the slope of XY. So the slope of the perpendicular bisector is:

Slope of perpendicular bisector = 1/7

Now we can use the point-slope form of a line to find the equation of the perpendicular bisector. Using the midpoint (6,10) and the slope 1/7, we get:

y - y1 = m(x - x1)

y - 10 = (1/7)(x - 6)

y - 10 = (1/7)x - 6/7

y = (1/7)x + 64/7

So the equation of the perpendicular bisector of segment XY is y = (1/7)x + 64/7.

Answer 2

E. x +7y = 4

F. x -7y = -64

Explanation:

You want the perpendicular bisectors of AB and XY, where the given points are ...

A(-2, 8) and B(-4, -6)
X(5, 17) and Y(7, 3)

Midpoint

The midpoint M of a segment RS is given by ...

M = (R +S)/2

The midpoint of AB is ((-2, 8) +(-4, -6))/2 = (-6, 2)/2 = (-3, 1).

The midpoint of XY is ((5, 17) +(7, 3))/2 = (12, 20)/2 = (6, 10).

Equation

The line perpendicular to the line through points (x1, y1) and (x2, y2) can be written as ...

(x2 -x1)(x -Mx) +(y2 -y1)(y -My) = 0 . . . . . . where M(Mx, My) is the midpoint

E. Segment AB

Using this form, we can write an equation for the bisector as ...

(-4 -(-2))(x -(-3)) +(-6 -8)(y -1) = 0

-2(x +3) -14(y -1) = 0 . . . . . . . simplify a bit

-2x -6 -14y +14 = 0 . . . . . . . eliminate parentheses

8 = 2x +14y . . . . . . . . . . . . . add 2x +14y to give positive x-coefficient

x +7y = 4 . . . . . . . . . . . . . . divide by 2 to get standard form

F. Segment XY

As above, and equation is ...

(7 -5)(x -6) +(3 -17)(y -10) = 0

2x -12 -14y +140 = 0 . . . . . . simplify

x -7y = -64 . . . . . . . . put in standard form

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Additional comment

We like "standard form" because the coefficients are integers. In this form, the leading coefficient is positive, and they are all mutually prime.

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