Name: Perpendicular Bisector Theorem Problems | 视频 | 干净世界
Uploaded: 2023-12-01T23:22:45.619Z
Duration: 7 min 23 s
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Solving Geometry problems using the Perpendicular Bisector Theorem 

The Perpendicular Bisector Theorem States
If a point is on the perpendicular bisector of a segment, then it is equidistant, or the same distance,from the endpoints of the segment

Transcript
Hi welcome to MooMooMath
Today we are going to learn about the Perpendicular Bisector Theorem
What is a perpendicular bisector? Let’s break it down
First, what is perpendicular?
Perpendicular means that you have two lines that intersect at a right angle.
We also have a bisector. 
A bisector cuts a segment into two equal parts right in the middle.
If a point is on the perpendicular bisector of a segment, then it is equal distant, or the same distance, from the endpoints of the segment
So we have GF as our segment
We have AB is our Perpendicular Bisector that means every single point that falls on this line is equal distance from F and G
So this point (A) is an equal distance from F and G because it falls on the perpendicular bisector.
This is nice in Geometry because it makes for two congruent triangles next to each other.
So given L the line is the perpendicular bisector of FG the segment, we can conclude that AF is equal to AG and those distances are equal to each other.
Now lets apply that perpendicular bisector theorem.
Our first diagram
We have TS is our segment and we have RV is a perpendicular bisector.
TV is 14 so we know this one is 14.
So TV is 14 so we know this one is 14 therefore R will be the same distance from s as it is from T so that will be 16
Lets look at the second diagram.
We are looking for the length of AB 
What do we know?
The segment AB is being bisected by the segment DC
We know this side is 2.5 so the whole segment will be 5 because we add those together.
And BC will be four because we have two congruent triangles, but that is not what the question was asking.
Now let's look an algebraic problem JK is our segment and L is our midpoint and JL and Lk are both congruent. HJ and HK and if they are both congruent then we can set up the algebraic expressions as equal, and then solve. 
3x + 1 = 5x-3 
Subtract 3x and that gives us 1 = 2x-3
Add 3 so 4  = 2x 
x = 2
So that is our answer, but we are not done. Let’s go back and look they are asking the length of HJ the segment. 
So we will take the fact that x =2 and plug it into our expression. 
3 x 2 +1 =7 so HJ is 7
Now let's go down and write some equations.
This next section says, “write an equation in point slope form for the perpendicular bisector of the segment with the given endpoints.
Let's visualize what we have.
We have a segment AB so we have two pints with a segment in between.
We want to find the equation of the perpendicular bisector segment.
What is the equation of the line that cuts it in half? 
The first step is to find where does this perpendicular bisector cross?
Let's find the midpoint of AB 
We need to know the midpoint formula. 
(x1 +x2 /2, y1+y2/2)
Add your two x coordinates together and divide by 2 which is 3
The y coordinate is -3 +5/2 = 1 so our midpoint is (3.1)
The second thing we need to find is we have to know what 
Is the slope of AB?
So the relationship between the line and its perpendicular is a negative reciprocal. Let's take our ordered pair and find slope.
Slope equation is y2-y1/x2-x1 = 5--3/0-6 = 8/-6,
which simplifies to -4/3 which is the slope ab.
We don't want the slope of AB we want the slope of its perpendicular.
-4/3 perpendicular slope is 3/4 and is +3/4
Take this new slope and that is perpendicular to AB and put it together with the midpoint in the midpoint formula.
I will replace my y with m which is my slope (x2-x1) 
so I will plug in midpoint for x1,y1 and my slope for M
This is x1,y1
so y-1 = 3/4(x-3),
and there is your equation of the perpendicular bisector.
Good luck with this and I hope this video was helpful on Perpendicular Bisector Theorem

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