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Learn it your way
A short class with Sarah and her students. The other ways are under More.
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Today, we are going to discuss circles. Can anyone tell me what a circle is?
A circle is a round shape with all points the same distance from the center!
Great! Exactly! A circle is a collection of all points in a plane that are at a constant distance, or radius, from a fixed point called the center. Now, what about the terms related to circles?
Terms like chord, tangent, and sector!
That's right! Let's move on to how lines can interact with a circle. Can anyone share the three ways?
A line can be a non-intersecting line, a secant, or a tangent!
Perfect! Remember: Non-intersecting means no contact, Secant means two points of intersection, and Tangent touches at one point. Keeping things simple with 'NST' helps us remember!
Now let’s summarize: Circles are defined by their center and radius, and when lines interact with circles, they can follow three pathways!
Let’s explore tangents to a circle. Who can tell me what a tangent is?
It’s a line that touches the circle at exactly one point.
Exactly! To see this in action, let’s go through Activity 1. Picture a straight wire touching a circular wire at a single point. When you rotate the wire, it only touches the circle at one point. What does that tell us?
That there’s only one tangent at any point on the circle!
Yes, wonderful! And what’s significant about the tangent’s relationship with the radius?
The tangent is always perpendicular to the radius at the point of contact!
Good job! So, our key takeaway today is: Tangents touch circles at one point and are perpendicular to the radius at that point!
Now, let's discuss how many tangents can be drawn from certain points. Can someone share what we found with regard to points inside a circle?
There are no tangents from points inside the circle!
Correct! What about points on the circle?
There’s exactly one tangent at a point on the circle!
Fantastic! And for points outside the circle?
You can draw two tangents from a point outside the circle!
Exactly! To remember, think of 'None, One, Two' depending on whether you're inside, on, or outside the circle. So 'N' for none, 'O' for one, and 'T' for two. Let’s summarize: From inside, no tangent; from on, one tangent; and from outside, two tangents!
Lines can be either non-intersecting, secants, or tangents when they interact with circles.
A tangent always touches a circle at exactly one point and is perpendicular to the radius at that point.
The number of tangents from a point is determined by its position relative to the circle: none, one, or two.
A tangent just can touch and go, it's perpendicular, this we know.
Imagine a tightrope walker touching the edge of a great circle, never falling in, just neatly gliding on the edge—like a tangent!
Use 'N,O,T': No tangents inside, One tangent on, Two tangents outside.
Remember 'TCR': Tangent touches, Circle radius, and interacts.
Term
What is a tangent?
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Term
What is the relationship between a tangent and a radius?
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A round shape in a plane with all points equidistant from the center.
The distance from the center of the circle to any point on its circumference.
A line that touches a circle at exactly one point and does not enter it.
A line that intersects a circle at two points.
The single point at which a tangent touches the circle.
This section explores the properties of circles, specifically focusing on the concept of tangents and their relationship to radius and secants.
In this section, we delve into the nature of circles, examining the definitions and properties of tangents, secants, and the various scenarios when a line interacts with a circle. Students will learn about the uniqueness of tangents, their perpendicular relationship with radii, and how to determine the number of tangents that can be drawn from different external points in relation to a circle.
In this section, we begin with a recap of what a circle is and the various terms associated with it, such as chords, segments, sectors, and arcs. We then focus on the interactions between lines and circles in planes, identifying three main scenarios:
The section explains the existence of tangents, detailing two activities to visualize the concept:
We also discuss important properties of tangents including:
Furthermore, the section covers the number of tangents from a point:
Theorem 10.2 emphasizes the equality of lengths of tangents drawn from an external point.
Finally, practical examples serve to illustrate the concepts, leading to exercises designed to reinforce student understanding.