Circles and Lines

 

By Pei-Chun Shih

 

 

 

1.  Central Angle Theorem:

The central angle subtended by two points on a circle is twice the inscribed angle subtended by those points.

 

      Definitions:

An inscribed angle: an angle formed by two chords in a circle that have a common endpoint. This common endpoint forms

     the vertex of the inscribed angle. For example, Angle BAC is an inscribed angle.

A central angle: an angle whose vertex is at the center of the circle. For example, Angle BOC is a central angle.

An intercepted arc: the arc, such as BLC, inside a central angle or inscribed angle is called the intercepted arc.

 

Click HERE to see the animation of point A.

 

 

      Proof of the theorem:

 

      Discussion about how to use technology to help students understand this theorem:

I would open the Geometer's Sketchpad file linked above and ask my students to click the animation bottom to observe the relationship of different inscribed angles and the central angle with the same intercepted arc.

 


 

2.  Inscribed Right Triangles:

(a) If a right triangle is inscribed in a circle, then its hypotenuse is a diameter of the circle.

(b) If one side of a triangle inscribed in a circle is a diameter of the circle, then the triangle is a right triangle and the   angle opposite the diameter is the right angle.

 

 

 

Click HERE to see the animation of point C.

 

 

      Proof of the theorem:

      Discussion about how to use technology to help students understand this theorem:

It might be a good idea to bring student to the computer lab and offer them opportunity to construct relations (a) and (b) by themselves.

 

 


 

3.  Intersecting Chord Theorem:

When two chords intersect each other inside a circle, the products of their segments are equal.

 

      Definitions:

A chord: a line that links two points on a circle or curve. It is a line segment that only covers the part inside the circle.

A diameter: a chord that passes through the center of the circle.

 

 

Let A, B, C, and D be points on a circle, with A not equal to B, and C not equal to D.

Suppose that the lines AB and CD intersect at a point P. Then (PA)(PB) = (PC)(PD)

 

Click HERE to see the animation of point A.

 

 

      Proof of the theorem:

 

      Discussion about how to use technology to help students understand this theorem:

I would open the Geometer's Sketchpad file linked above and ask my students to click the animation bottom to observe that PA*PB is always equal to PC*PD no matter where the chords are.

 


 

4.  Length of an Arc:

The length of any circular arc s is equal to the product of the measure of the radius of he circle r and the radian measure of the central angle q that it subtends.

 

      Relationship between a whole circle and a circular arc:

 

 

 

Click HERE to see the change of the arc length in different angle measures.

 

 

      Proof of the relationship:

Here, the circumference of a circle is an arc length of the whole circle. By definition, p is the ratio of the circumference of a circle to the diameter which equals two times of the radius, r. Therefore, p = circumference/2r.

In the circle above, the arc length S is a portion of the circumference of the circle. Therefore,

 

      Discussion about how to use technology to help students understand this relationship:

Again, using the Geometer's Sketchpad file to show different arc lengths with different central angles is a good way to help students to understand this concept. I also try to use some real-life examples, such as using a disk to represent a whole circle and a piece of pie to represent a portion of circle, to enhance students understanding of the proof stated above.

 

 

 

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