Circle At Origin Does (8, √17) Lie On It? Explained
Hey guys! Let's dive into a cool math problem about circles and points. We're going to figure out if a point lies on a circle, and we'll break it down step by step so it's super clear. So, grab your thinking caps, and let's get started!
Understanding the Problem
So, here's the deal: we have a circle that's centered right at the origin – that's the point (0, 0) on our coordinate plane. This circle has a special guest, the point (0, -9), which sits right on its edge. Now, the big question is: does another point, (8, √17), also hang out on this circle? To figure this out, we need to understand a bit about circles and how distances work. Essentially, we need to find the radius of the circle using the given point and then check if the distance from the origin to the other point matches this radius. If it does, then that point is also on the circle; if not, it's somewhere else!
Finding the Radius of the Circle
First things first, let's nail down the radius of our circle. Remember, the radius is the distance from the center of the circle to any point on its edge. We know the center is at (0, 0), and we have a point on the circle: (0, -9). To find the distance between these points, we can use the distance formula. The distance formula is derived from the Pythagorean theorem, and it helps us calculate the straight-line distance between two points in a coordinate plane. It looks like this:
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Where (x₁, y₁) and (x₂, y₂) are the coordinates of our two points. In our case, (x₁, y₁) is (0, 0), and (x₂, y₂) is (0, -9). Let's plug these values into the formula:
Distance = √[(0 - 0)² + (-9 - 0)²] Distance = √[0² + (-9)²] Distance = √[0 + 81] Distance = √81 Distance = 9
So, the radius of our circle is 9 units. This means every point on the circle is exactly 9 units away from the origin. Keep this number in mind – it's our benchmark!
Checking the Distance to the Second Point (8, √17)
Alright, now we've got the radius sorted out, let's see if the point (8, √17) is also chilling on our circle. We're going to use the distance formula again, but this time, our points are (0, 0) (the center) and (8, √17). Let's plug those numbers in:
Distance = √[(8 - 0)² + (√17 - 0)²] Distance = √[8² + (√17)²] Distance = √[64 + 17] Distance = √81 Distance = 9
Guess what? The distance from the origin to the point (8, √17) is also 9 units! That's the same as our radius. This is super important because it confirms whether the point lies on the circle or not. We've done the math, and it's looking like our second point might just be a part of the circle crew.
Conclusion: Does (8, √17) Lie on the Circle?
So, let's bring it all together. We found that the distance from the origin (0, 0) to the point (8, √17) is 9 units, which is exactly the same as the radius of the circle. What does this mean? It means that the point (8, √17) does indeed lie on the circle! Remember, any point that is the same distance from the center as the radius is on the circle. This is a fundamental concept in geometry, and we've just seen it in action.
Therefore, the answer is not A because the distances are actually the same, and it's not B because the radius isn't 10 units; it's 9 units. The point (8, √17) lies perfectly on the circle. Great job, guys! We tackled this problem like pros, using the distance formula and our understanding of circles to solve the mystery.
Why This Matters: Real-World Applications and Further Exploration
Understanding circles and the distance formula isn't just about acing math problems; it has some pretty cool real-world applications too! For example, think about GPS systems. They use the concept of distances from satellites (which can be thought of as points in space) to your location to pinpoint where you are on Earth. The principles we used to solve this problem are similar to the calculations that make GPS work.
Circles are also fundamental in engineering and architecture. When designing structures or mechanical systems, engineers need to understand the properties of circles to ensure everything fits together and functions correctly. From the wheels on a car to the gears in a machine, circles are everywhere!
If you're curious to explore more, you can dive into the equation of a circle, which gives you a way to describe any circle mathematically. The standard form of a circle's equation is: (x - h)² + (y - k)² = r², where (h, k) is the center of the circle, and r is the radius. Try plugging in the values from our problem and see how it all fits together! You can also investigate other geometric shapes and their properties. Geometry is a fascinating field with endless possibilities for exploration and discovery.
Further Practice: Applying the Concepts
Want to really nail this concept? Here are a couple of practice problems you can try:
- A circle is centered at the origin and passes through the point (5, -12). Does the point (-15, 8) lie on the circle? Show your work!
- A circle with a radius of 7 is centered at (2, 3). Does the point (9, 3) lie on this circle? Explain your reasoning.
Working through these problems will help you solidify your understanding of circles, radii, and the distance formula. Remember, practice makes perfect! And if you get stuck, don't be afraid to review the steps we've covered in this article or ask for help. Math is a journey, and every problem you solve brings you closer to mastering the concepts.
Repair Input Keyword
Original Question: A circle centered at the origin contains the point (0, -9). Does (8, √17) also lie on the circle? Explain.
Rewritten Question: A circle is centered at the origin and includes the point (0, -9). Does the point (8, √17) also lie on this circle? Please provide an explanation.
SEO Title: Circle at Origin Does (8, √17) Lie on It? Explained