# Perpendicular bisector question and general checking

• greener1993
In summary, the perpendicular bisector of a straight line joining the points (3,2) and (5,6) meets the x-axis at A and the y at B. The distance AB is equal to 6root5.
greener1993
The main help i need is with this question:

The perpendicular bisector of a straight line joining the points (3,2) and (5,6) meets the x-axis at A and the y at B. Prove that the distance AB is equal to 6root5.

I just need to have theses checked to make sure there right, would love if you could just skim it and check :D

1)What is the gradient of the line perpendicular to y=3x-5. hence find the equation of the line which goes through (3,1) and is perpendicular to y=3x-5
a) to find the gradient I did, m2=-1/m1. m1=3 so it was -1/3 = -1/3
b) We know the coordinates (3,1) which is the y and X, so 1=(-1/3*3)+c
1=-1+c, so
2=C all together the answer is y=-1/3x+2

2)Find the equation of line B which is perpendicular to line A and goes though the mid-point.
I am given the coordinates for line A, (2,6) and (4,8). Using that i can find both the midpoint and the gradient of line A = midpoint = (3,7) Gradient = 1
So M2=1/M1 so -1/1=-1
we know from midpoint, y=7 and x=3 so
7=(-1*3)+c
7+3=c
10=c so the equation is Y=-x+10

greener1993 said:
The main help i need is with this question:

The perpendicular bisector of a straight line joining the points (3,2) and (5,6) meets the x-axis at A and the y at B. Prove that the distance AB is equal to 6root5.
I presume, from the problems below that you know that the straight line joining (3, 2) and (5, 6) has slope (6- 2)/(6-3)= 4/2= 2 and that the midpoint is ((3+ 5)/2, (2+ 6)/2)= (4, 4).
What is the line with slope -1/2 that passes through (4, 4)?

The point "A" is (x, 0) and the point "B" is at (0, y). Knowing the equation of the line, it should be easy to find x such that y= 0 and find y such that x= 0.

The distance AB is equal to $\sqrt{x^2+ y^2}$ where x and y were found above.

I just need to have theses checked to make sure there right, would love if you could just skim it and check :D

1)What is the gradient of the line perpendicular to y=3x-5. hence find the equation of the line which goes through (3,1) and is perpendicular to y=3x-5
a) to find the gradient I did, m2=-1/m1. m1=3 so it was -1/3 = -1/3
b) We know the coordinates (3,1) which is the y and X, so 1=(-1/3*3)+c
1=-1+c, so
2=C all together the answer is y=-1/3x+2

2)Find the equation of line B which is perpendicular to line A and goes though the mid-point.
I am given the coordinates for line A, (2,6) and (4,8). Using that i can find both the midpoint and the gradient of line A = midpoint = (3,7) Gradient = 1
So M2=1/M1 so -1/1=-1
we know from midpoint, y=7 and x=3 so
7=(-1*3)+c
7+3=c
10=c so the equation is Y=-x+10
Yes, both of those are correct.

:D thank you for checking them two questions for me.

As for the main one, putting into an equation had sliped me, mainly because i don't know how to find the x intecepy from a y=mx+c graph... However i put into equation and got -1/2x +8, so the y intercept is 8. but was unsure from there for the x axis. Working back however, knowing the answer is 6root5 or root36*root5, we know the answer must be 180 before rooting. 8*8 =64 180-64 = 116 and root116 isn't a whole number, something my teacher is unlikly to do. so I am not even confident i know how to find the y :S

ouch some very bad maths sorry, -1/2 *4 = 2 :P there for c=6 :D making what i said ilrelivant

## 1. What is a perpendicular bisector?

A perpendicular bisector is a line or line segment that intersects another line at a 90 degree angle and divides it into two equal parts.

## 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 slope of the given line. Then, use the negative reciprocal of that slope to find the slope of the perpendicular bisector. Finally, use the point-slope formula or the midpoint formula to find the equation of the perpendicular bisector.

## 3. What is the relationship between a perpendicular bisector and the midpoint of a line segment?

The perpendicular bisector of a line segment always passes through the midpoint of that line segment. This means that the point where the perpendicular bisector intersects the given line segment is exactly halfway between the two endpoints of the line segment.

## 4. How can you check if a point lies on a perpendicular bisector?

To check if a point lies on a perpendicular bisector, you can plug in the coordinates of the point into the equation of the perpendicular bisector. If the coordinates satisfy the equation, then the point lies on the perpendicular bisector. Another way to check is to measure the distance from the point to the two endpoints of the line segment. If the distances are equal, then the point lies on the perpendicular bisector.

## 5. What is the purpose of checking if a line is a perpendicular bisector?

Checking if a line is a perpendicular bisector is important because it allows us to determine if a line segment is divided into two equal parts. This is useful in many geometric and algebraic problems, such as finding the center of a circle inscribed in a triangle or finding the shortest distance from a point to a line.

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