00:01
Okay, so in this problem, there's three things we want to prove.
00:03
So i'm going to do it three different times.
00:05
So here's the first one.
00:07
I'm going to prove that if a diameter of a circle bisects a cord, that's not a diameter, then the diameter is perpendicular to the cord.
00:15
Okay.
00:17
So here's what we have.
00:18
We have a circle.
00:21
And we have some cord.
00:23
And then we're saying that this diameter here is bisecting the cord.
00:31
So we're given, this is ab and i'll call the chord cd, given ab is a diameter.
00:46
It's the first thing.
00:51
And then we're also given that ab bisects cd.
01:02
Okay.
01:04
And let's just say they hit at point m right there.
01:06
Okay, we want to prove, we want to prove that a.
01:17
Ab is perpendicular to cd.
01:22
All right.
01:23
So, you know, standard two column proof i'll do here.
01:27
So start off with what we're given.
01:31
We're given ab as a diameter.
01:34
Cool thing about a diameter is it goes through the center.
01:37
So ab is a diameter.
01:45
That's given.
01:50
A .b.
01:51
Goes through the center.
02:00
I feel i'm going to run around.
02:01
That's definition of diameter.
02:02
Goes to the center.
02:09
Okay.
02:11
And we'll call the center o just because we know that a .b.
02:17
Is a bisector.
02:18
C .d.
02:25
It's given.
02:28
So that means cm is congruent to md.
02:33
A definition of a bisect.
02:42
Okay.
02:50
Bissect.
02:52
Okay.
02:53
Now when we construct.
02:57
If we construct o .d.
03:02
And o .c.
03:04
From the center, we're assuming o is the center.
03:08
The diameter has the center somewhere on the diameter.
03:10
Just find it.
03:11
Construct o .c.
03:12
And o .d.
03:14
We can say o .c is congruent to o .d because they're both radii.
03:24
All radii are congruent.
03:28
Okay.
03:30
So notice that that gives us a.
03:31
Side of a triangle up above where we said from the definition of bisector gives us another side and now we can say that om is congruent to itself still called the reflexive property run that room so i'll make it blue this side right here it's part of the triangle on the left also part of the trying on the right so that gives us a third side.
04:08
So now we know the two triangles are congruent so we can say i'll kind of continue my proof over here.
04:16
Triangle o -c -m is congruent to triangle o -d -m because of the side -side -side -side -congruance theorem.
04:33
Since they are congruent we can say that angle o -m -c is congruent to angle o -m -d and because they're corresponding parts of congruent triangles, so they're congruent.
04:50
So now we know that they are 90 degrees because since they are adjacent angles that are congruent, then they have to be 90 degrees.
04:59
So angle omc, the measure of angle omc equals the measure of angle omd, equals 90 degrees because they're adjacent and congruent angles.
05:31
And adjacent and congruent angles have to be 90 degrees.
05:35
So since the ab and cd hit and form 90 degree angles, right angles, we can now say ab is perpendicular to cd because they mean.
05:55
At right angles also sometimes called the definition of being perpendicular degrees okay so that's how you would do that's how you do that's part a all right now we got to go to part b the second part and it says it's just it's a different given so that was the first one now it says also prove that the perpendicular bisector of a cord goes through the center of a circle.
06:22
Okay.
06:23
So it's just a different given.
06:25
Here's our circle.
06:27
So here's this cord, cd, and we're going to get the perpendicular bisector.
06:35
So here it is right here.
06:37
I'll call it ab again.
06:40
Okay.
06:42
So we know now we're given this time that ab, ab is the perpendicular bisector.
07:08
Of cd.
07:10
That's what we're given.
07:12
We want to prove that ab goes through the center.
07:24
Through center of circle.
07:32
All right.
07:32
So i'll start my proof kind of below where i wrote everything.
07:35
So start with what you're given.
07:37
Again, we'll call this point m, right? we know that.
07:44
So start with what you're given.
07:46
I'm going to skip that and jump down to what we could say from what is given.
07:49
Cm is congruent to md because it's a bisector...