Unit 7, Lesson 1
Connecting Distance and Circles
Integrated Math 2
1.1
Activity
Plot as many points as you can that are a distance of 5 away from the point .
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Plot as many points as you can that are a distance of 5 away from the point .
Here is a list of points.
The image shows a circle with its center at and radius of 17 units.
Let's find some points that are a distance of 5 away from the point and plot them on a grid. We can use gridlines to find 4 points:
We can use the Pythagorean Theorem to help us find some more points. Recall that for a right triangle with a hypotenuse of 5, we can make a right triangle with legs 3 and 4. This means that if the horizontal and vertical distances between the points are 3 and 4, then the distance between them will be 5. We can find 8 more points this way!
As we fill in more points, it looks like the points are forming a circle around the point , and in fact, they are. The circle has a radius of 5, which makes sense since we know that a circle is the set of all points a given distance away from a central point. In this case, that given distance is 5.
How can we check whether a point lies on this circle if we are not sure? Let's check the point . We can use the Pythagorean Theorem to test the distance between the center of the circle and the point: , which is not equal to 5. That means the point is close to the circle, but does not lie on the circle.
We can say that for a circle of radius and center , if a point is a distance of away from , it will lie on the circle. If point is not a distance of away from , then it will not lie on the circle. Another way to say this is that for a circle of radius and center , will lie on the circle if and only if it is a distance of away from .