Equation of Perpendicular Bisector

In summary, the correct equation of the perpendicular bisector of the segment joining the points (7,0) and (1,8) is -3x + 4y = 4. This is obtained by finding the negative reciprocal slope of the line joining the two points, using the midpoint formula to find a point on the perpendicular line, and then using the point-slope form to find the final equation. The discrepancy between this answer and the one given in the book is likely a textbook error.
  • #1
TbbZz
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Homework Statement



Write an equation of the perpendicular bisector of the segment joining the points (7,0) and (1,8).

Homework Equations



Midpoint formula, Perpendicular slope (negative reciprocal of a line is the slope of a line perpendicular to the first line)

The Attempt at a Solution



First I get the slope of the line:
m = (8-0)/(1-7) = 8/-6 = -4/3

Then I take the negative reciprocal of it:
m[perpendicular line] = 3/4

Then I use the midpoint formula between the two given points, to find a point on the perpendicular line.
midpoint = (1+7)/2, (8+0)/2
= (4,4)

so I now have the line y=(3/4)x + b as the line. I plug in 4,4
4 = 3/4(4) + b
I solve b to be 1 (b = 1)

so now I have y = 3/4 (x) + 1 as the line.

I'm supposed to give the answer in standard form, so I do:
m =-A/B = 3/4 to get
A = -3
B = 4

and b = C/B = 1 to get
b = 1 = C/4
so C = 4

So my final answer is -3x + 4y = 4
However the correct answer in the back of the book is -3x + 4y = -4

What am I doing wrong?
Thanks for reading.
 
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  • #2
Your work is correct for the points you supplied. It is probably a textbook error.
 
  • #3
your slope calculation is wrong in the beginning :)
 
  • #4
sarahmaliha said:
your slope calculation is wrong in the beginning :)

no it isnt. he's done it right. it must be a textbook error.
 
  • #5
Certainly the midpoint is (4, 4) and (4, 4) satisfies -3x+ 4y= 4, not -3x+ 4y= -4.
 

1. What is the equation of a perpendicular bisector?

The equation of a perpendicular bisector is a line that passes through the midpoint of a segment and is perpendicular to that segment. It can be represented in the form of y = mx + b, where m is the slope of the line and b is the y-intercept.

2. How do you find the equation of a perpendicular bisector?

To find the equation of a perpendicular bisector, you need to first find the midpoint of the segment using the formula (x1+x2)/2, (y1+y2)/2. Then, find the slope of the segment using the formula (y2-y1)/(x2-x1). Finally, use the negative reciprocal of the slope and the midpoint coordinates to write the equation in the form of y = mx + b.

3. Can the equation of a perpendicular bisector be written in any form?

Yes, the equation of a perpendicular bisector can also be written in the general form of ax + by + c = 0, where a, b, and c are constants. This form is useful when working with equations of multiple lines and finding intersections.

4. What is the significance of the equation of a perpendicular bisector?

The equation of a perpendicular bisector is significant because it helps us find the midpoint of a segment and determine if two segments are perpendicular to each other. It is also useful in geometry and trigonometry when solving problems involving triangles and circles.

5. Can the equation of a perpendicular bisector apply to non-linear segments?

No, the equation of a perpendicular bisector only applies to straight line segments. Non-linear segments, such as curves, do not have a constant slope and therefore cannot have an equation in the form of y = mx + b.

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