yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Proof: perpendicular radius bisects chord


2m read
·Nov 10, 2024

So we have this circle called circle O based on the point at its center, and we have the segment OD, and we're told that segment OD is a radius of circle O. Fair enough! We're also told that segment OD is perpendicular to this chord, to chord AC, or to segment AC.

What we want to prove is that segment OD bisects AC. So another way to think about it is that it intersects AC at AC's midpoint.

Pause this video and see if you can have a go at that.

All right! Now let's go through this together, and the way that I'm going to do this is by establishing two congruent triangles. Let me draw these triangles. So I'm going to draw one radius going from O to C and another from A to O.

Now we know that the length AO is equal to OC because AO and OC are both radii in a circle. The length of the radius does not change, so I can put that right over there.

And then we also know that OM is going to be congruent to itself. It's a side in both of these triangles, so let me just write it this way: OM is going to be congruent to OM. This is reflexivity. Reflex, kind of obvious. It's going to be equal to itself—it's going to be congruent to itself.

So you have it just like that! And now we have two right triangles.

How do I know they're right triangles? Well, they told us that segment OD is perpendicular to segment AC and our assumptions in our given. If you just had two triangles that had two pairs of congruent sides, that is not enough to establish congruency of the triangles.

But if you're dealing with two right triangles, then it is enough!

And there's two ways to think about it. We had thought about the RSH postulate where if you have a right triangle and, or two right triangles, you have a pair of sides are congruent and the hypotenuses are congruent, that means that the two triangles are congruent.

But another way to think about it, which is a little bit of common sense, is using the Pythagorean theorem. If you know two sides of a right triangle, the Pythagorean theorem would tell us that you could determine what the other side is.

And so what we could say is—and let's just use RSH for now—but you could also say we can use the Pythagorean theorem to establish that AM is going to be congruent to MC.

But let me just write it this way: I will write that triangle AMO is congruent to triangle CMO by RSH.

And if the triangles are congruent, then the corresponding sides must be congruent. So therefore we know that AM—segment AM—is going to be, I'm having trouble writing, congruent to segment CM.

That these are going to have the same measure! And if they have the same measure, we have just shown that M is the midpoint of AC or that OD bisects AC.

So let me just write it that way: therefore, OD bisects AC. Segment OD bisects segment AC, and we're done!

More Articles

View All
The Insane Math Of Knot Theory
Most of us tie our shoelaces wrong. There are two ways to tie a knot in your shoelaces. In one, you go counterclockwise around the loop, and in the other, you go clockwise. These two methods look almost identical, but one of these knots is far superior to…
It Started: My Thoughts On The Joe Biden Tax Plan
What’s up guys? It’s Graham here. So normally, I don’t make videos like this, and I try to stay away from topics that could be taken out of context or politicized. But lately, there’s been a lot of talk about a brand new tax plan that would soon increase …
More Compute Power Doesn’t Produce AGI
The artificial general intelligence crew gets this completely wrong too. Just add more compute power and you’ll get intelligence when we don’t really know what it is underneath that makes us creative and allows us to come up with good explanations. Peopl…
Would You Bite Into a Raw Bison Liver? | Expedition Raw
What am I supposed to do with this? I am a white kid from Brooklyn, and I’d never been to an Indian Reservation before. Is this the liver? Take a bite. The bite like this. I’ll never… just like this. You all have to now. It’s completely sterile. There you…
Jordan Peterson | You Have No More Time
You need a family. You need friends. You don’t need to have all these things, but you better have most of them: family, friends, career, educational goals, plans for, you know, time outside of work, attention to your mental and physical health, etc. You k…
How I Choose Opportunities That Align With My Brand | Behind The Velvet Rope PT 2
The social media following and the celebrity makes my deal making better because I could deliver the company’s extraordinary reductions in customer acquisition costs. You can’t run around all day long worrying what people think about you. I don’t. You sho…