[G.C.2] Circles #2

Objective

Common Core Text:

• [G.C.2] Identify and describe relationships among inscribed angles, radii, and chords. Include

  • the relationship between central, inscribed, and circumscribed angles;

  • inscribed angles on a diameter are right angles;

  • the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Said Differently:

• Identify and describe relationships among inscribed angles, radii, and chords. Include

  • • Tangent-Radius Theorem

• • Inscribed Angle Theorem

  • Circumscribed Angle Theorem

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Explanation

There are a lot of parts of a circle, and they are all related to each other. Before we look at their relations, let's review the parts

Radius, Diameter, Chord, Circumference

You learned these in previous grades. If you've forgotten, be sure to review them. You can check them out here or any other way you prefer

Arc, Minor Arc, Major Arc.

Arc is one of those words that gets used differently by different people. Best to just understand the idea, and use your own judgement as to what they are talking about.

Regard the following picture

If someone was talking about , you might think they mean the red line between C and D. But they could actually be talking about the blue line, because in a circle's world, that line is also between C and D. So how do you know which line someone is talking about? They need to tell you if they're talking the minor arc or the major arc.

Here are 3 equivalent statements. The minor arc is the arc

• whose subtending angle is less than 180°

• said differently, the smaller of the two arcs

• said differently, whose length is less than half of the circumference

In this picture you would write

• minor

Here are 3 equivalent statements. The major arc is the arc

• whose subtending angle is greater than 180°

• said differently, the larger of the two arcs

• said differently, whose length is more than half of the circumference

In this picture you could say

• major

• or, because you have point E that is part of the major arc CD, you could say

Tangent Lines

Tangent lines are cool but can be a little hard to truly wrap one's mind around. Essentially, tangent lines are lines that intersect the circumference of a circle at exactly 1 point.

In this picture, line FG intersects the circle at point G, and only point G. Therefore we can say the following statements (all of these are different ways of saying the same thing)

• • Line FG is tangent to the circle

  • Line FG is tangent

  • Line FG is a tangent line

  • Line FG is a tangent

Looking at line segment AB, one might say "Well, line segment AB only intersects the circle at point A, so it's tangent, right?" Wrong. When dealing with line segments, you have to imagine them as lines. You can see if we treated line segment AB as a line instead of a line segment, it would intersect the circle a second time at the bottom. That means it's not tangent. Therefore

• • Line segment AB is not tangent to the circle

  • Line segment AB is not tangent

  • Line segment AB is not a tangent line

  • Line segment AB is not a tangent

Central Angles, Inscribed Angles, and Circumscribed Angle

There are 3 types of angles that circles can have.

Central Angles

• Vertex is the center

• Sides are radii of the circle

• In this picture, ∠CAD is a central angle

Inscribed Angles

• Vertex lies on the circumference

• Sides are chords of the circle

• In this picture, angle ∠FBE is an inscribed angle

Circumscribed Angles

• Vertex is outside of the circle

• Sides are tangent to the circle

• In this picture, angle ∠GKH is a circumscribed angle

Relationships Within Circles

• Tangent-Radius Theorem

• Inscribed Angle Theorem

• Circumscribed Angle Theorem

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Practice your skills