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# ZingPath: Trigonometric Ratios and Circles

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## Trigonometric Ratios and Circles

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### Lesson Focus

#### Arcs and Angles of a Circle

Geometry

Observe the relationship between the arcs and angles of a circle.

### Now You Know

After completing this tutorial, you will be able to complete the following:

• Explain the relationship between central angles and intercepted arcs.
• Explain the relationship between inscribed angles and intercepted arcs.
• Explain the relationship between central angles and inscribed angles.

### Everything You'll Have Covered

One Important branch of mathematics is geometry. Geometry is concerned with the properties of configurations of geometric objects, such as points, lines, and circles. It is important to be familiar with the following key vocabulary terms before beginning this Activity Object:

• angle -a figure formed by two noncollinear rays that share the same endpoint For example, In the figure shown, angle AOB is formed by the rays OA and OB with a common endpoint O.
• arc - a curved line that lies on the circumference of a circle For example, In the figure, the part of the circumference of the circle between points A and B is an arc.
• central angle - the angle in a circle whose vertex is the center of the circle For example, In the figure shown, O is the center of the circle and angle POQ is the central angle.
• circle -a closed curve in a plane with all its points equidistant from a given point called center For example, In the figure shown, O is the center of the circle. All points on the black ring, such as A, B, C, and D, are equidistant from O, the center. The length of the black ring is the circumference of the circle.
• diameter - a line segment whose endpoints are on the circle and which passes through the center of the circle For example, In the circle shown, the line segments AB and PQ are the diameters of the circle. Notice that they have their endpoints on the circle and that both of them pass through the center O of the circle.
• inscribed angle - the angle formed by two chords that meet at the same point on a circle For example, In the figure, AB and BC are two chords which meet at point B on the circle. So, angle ABC is an inscribed angle.
• intercepted arc - the part of a circle which lies between two lines that intersect the circle For example, In the circle below, the arc between A and B is the arc intercepted by the angle AOB.

Using this Activity Object,students will learn the following information:

1. The measure of a central angle ( ) is equal to the degree measure of its intercepted arc (AC).

2. AC is a diameter while the measure of the central angle subtended by any arc is equal to 180? and the measure of the inscribed angle on that arc is equal to 90?. With AC as diameter, subtended by any

3. The measure of an inscribed angle is equal to one-half the degree measure of its intercepted arc.

4. The measure of a central angle subtended by any arc is twice any of the inscribed angles on that arc.

### Tutorial Details

 Approximate Time 15 Minutes Pre-requisite Concepts arcs, measuring angles, parts of a circle Course Geometry Type of Tutorial Dynamic Modeling Key Vocabulary angles, arcs, circle