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

Points inside/outside/on a circle | Mathematics I | High School Math | Khan Academy


2m read
·Nov 11, 2024

A circle is centered at the point C which has the coordinates -1, -3 and has a radius of six. Where does the point P, which has the coordinates -6, -6, lie? We have three options: inside the circle, on the circle, or outside the circle.

The key realization here is just what a circle is all about. If we have the point C, which is the center of a circle, a circle of radius six. So, let me draw that radius. Let's say that is its radius; it is six units. The circle will look something like this. Remember, the circle is a set of all points that are exactly six units away from that center.

So, that's the definition of a circle; it's the set of all points that are exactly six units away from the center. If, for example, P is less than six units away, it's going to be inside the circle. If it's exactly six units away, it's going to be on the circle, and if it's more than six units away, it's going to be outside of the circle.

The key is, let's find the distance between these two points. If the distance is less than six, we are inside; if the distance equals six, we're on the circle; and if the distance is more than six, we are outside of the circle. So, let's do that.

If we wanted to find—and there are different notations for the distance—well, I'll just write D, or I could write the distance between C and P is going to be equal to—and the distance formula comes straight out of the Pythagorean theorem—but it's going to be the square root of our change in x² plus our change in y².

So what is our change in X? If we view C as our starting point and P as our endpoint (but we could do it either way), our change in X is -6 minus -1, so -6 minus -1 and we're going to square it. So what we have inside here—that is the change in X.

So, we're taking our change in X and then plus our change in y². So, we are going from -3 to -6, so our change in y is -6 minus -3, which is -6 - -3, and we're going to square everything. So, that is our change in y inside the parenthesis, and we're going to square it.

This is equal to -6 plus 1 is one way to think about it, so this is -5² and then this is -6 + 3, so that is -3². Once again, you can see our change in x is 5; we go five lower in X, and we're going three lower in y, so our change in y is 3.

So this is equal to the square root of 25 plus 9. The square root of 25 + 9, which is equal to the square root of 34. Now the key is, is the square root of 34 less than 6, greater than 6, or equal to 6?

Well, we know that 6 is equal to the square root of 36. So, the square root of 34 is less than the square root of 36. I could write the square root of 34 is less than the square root of 36, and so the square root of 34 is less than 6.

Since the distance between C and P is less than six, we are going to be on the inside of the circle. If I somehow got square root of 36 here, then we’d be on the circle, and if I somehow got square root of 37 here or something larger, we would have been outside the circle.

More Articles

View All
Complex numbers with the same modulus (absolute value)
[Instructor] We are asked, which of these complex numbers has a modulus of 13? And just as a bit of a hint, when we’re talking about the modulus of a complex number, we’re really just talking about its absolute value. Or if we were to plot it in the compl…
Why Stupid People Get Lucky?
Statistically, your odds of winning the lottery are one in 292 million. This means you’ve got a 0.0000338 chance of winning the Powerball jackpot. To put this into perspective, you’ve got a one in one million two hundred and twenty-two thousand chance of …
Graphs of rational functions: horizontal asymptote | Algebra II | High School Math | Khan Academy
Let f of x equal negative x squared plus a x plus b over x squared plus c x plus d, where a, b, c, and d are unknown constants. Which of the following is a possible graph of y is equal to f of x? They tell us dashed lines indicate asymptotes. So, this is…
Introduction to the Vedic Period | World History | Khan Academy
First civilization that we have evidence of around modern-day India and Pakistan is the Indus Valley Civilisation. It’s right around the Indus River in modern-day Pakistan and Northwest India. In other videos, we talked about how it really comes into bein…
Net exports and capital outflows
Let’s take a look at our GDP equation for an open economy. So, GDP is equal to national income, and that’s going to be equal to consumption plus investment plus government spending. And since this is an open economy, plus net exports. Now, the first thi…
Eventually You Will Get What You Deserve
We’re still talking about working for the long term. The next tweet on that topic is: apply specific knowledge with leverage, and eventually you will get what you deserve. I would also add to that: apply judgment, apply accountability, and apply the skill…