Proof: perpendicular radius bisects chord

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!