Finding arc measures | Mathematics II | High School Math | Khan Academy (2023)

Introduction

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Watch Sal solve a few problems where he finds a missing arc measure.

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Content

So I have some example questions here from khan academy on arc measure and like always, I encourage you to pause the video after you see each of these questions and try to solve them before I do so.

This first question says: what is the arc measure in degrees of arc ac on circle p below? So this is point a that is point c and when they're talking about arc a c, since they only have two letters here, we can assume this is going to be the minor arc.

When we talk about the minor arc, there's two potential arcs that connect point a and point c, there's a one here on the left and then there's the one on there is the one on the right and since c, isn't exactly isn't exactly straight down from a it's a little bit to the right.

The shorter arc, the arc with a smaller length or the minor arc is going to be this one that I'm depicting here right on the right.

So what is this arc measure going to be well? The measure of this arc is going to be exactly the same thing as in degrees as the measure of the central angle that intercepts the arc so that central angle, let me do it in a different color I'll.

Do it in this blue color.

That central angle is angle, c, p, a angle c p a and the measure of that central angle is going to be 70 degrees, plus 104 degrees.

It's going to be this whole thing right over there, so it's going to be 174 degrees, 174 degrees, that's the arc measure in degrees of arc ac.

Let's keep doing these, so let me do another one, so this next one asks us in the figure below in the figure below segment a d.

So this is point a this is point d, so segment, a d.

Is this one right over here? Let me see if I can draw that that's a d right over there, a d and c e are diameters of the circle.

So let me draw c e, so c e is we're going to connect point c and e.

These are diameters.

So let me so they go straight whoops, I'm using the wrong tool.

Let me so those are somehow.

I should all right all right, so those are diet whoops.

How did that happen? So let me somehow my pen got really big all right.

Let me almost there.

Okay, so c e.

There you go so those are both diameters of the circle p.

What is the arc measure of a b of arc, a b in degrees so arc a b once again, there's two potential arcs that connect point a and b there's the minor arc, and since this only has two letters, we'll assume it's the minor arc.

It's going to be this one over here, there's a major arc, but to denote the major arc, they would have said something like a e b or a d b or arc a cb to make us go this kind of the this long way around, but this is arc a b, so we in order to find the arc measure, we just really have to find the measure of the central angle.

This is the central angle that intercepts that arc, or you could even say it defines that arc in some way.

So how can we figure out this angle and this one's a little bit trickier? Well, the key to the key here is to realize that this 93 degree angle.

It is vertical to this whole angle right over here, and we know from geometry, which we're still learning as we do.

This example problem that vertical angles are going to have the same measure.

So if this one on this one is 93 degrees, and this entire blue one right over here is also going to be.

Let me write it.

This is also going to be 93 degrees, so 93 degrees.

That's going to be made up of this red angle that we care about and the 38 degrees.

So this red one, which is, which is the measure of the central angle.

It's also the arc measure of arc a b is going to be 93, minus 93 degrees, minus 38 degrees.

So what is that going to be? Let's see 93, I can write degrees there, minus 38 degrees that is going to be equal to.

Let's see if it was 93 minus 40, it would be, it would be 53.

it's going to be two more.

It's going to be 55 degrees, 55 degrees, and we are done this angle right over here is 55 degrees.

If you were to add this angle, measure plus 38 degrees, you'd get 93 degrees, and that has the same measure, because it's vertical with this angle right over here with angle d, p e, all right- let's do one more of these, so we have in the figure below and it doesn't quite fit on the page but we'll scroll down in a second a b is the diameter of circle p? Is the diameter of circle p all right, so a b is a diameter.

Let me label that, so a b is a diameter, so it's going straight across straight across the circle.

What is the arc measure of a b c in degrees, so a b c, so they're making us go the long way around.

This is a major arc they're talking about.

Let me draw it arc a what is the arc measure of arc a b c, so we're going the long way around, so it's a major arc.

So what is that going to be? Well, it's going to be in degrees, the same measure as the angle as the central angle that intercepts it.

So it's going to be the same thing as this central angle right over here.

Well, what is that central angle going to be well since we know that this is a diameter, since a b is a diameter, we know.

We know that this part of it, this part of it, is going to be 180 degrees, we're going halfway around the circle, 180 degrees, and so, if we want to look at this whole angle, the angle that intercepts the major arc abc is going to be the 180 degrees plus 69 degrees, so we're going to have 180 degrees plus 69 degrees, which is equal to what is that 249 249 degrees.

That's the arc measure of this major arc c.