Grade 9 Math – Circle Properties

Mathematical ideas related to the lesson

In order to complete this discovery activity, there are several mathematical terms that you must be familiar with. Please complete the “Activate your knowledge” section before proceeding with the activity.

Activate your knowledge

Circle Geometry Glossary

Complete this glossary by providing a definition for each term and completing each part of the illustration. Definitions can be found in Chapter 9 of your textbook as well as at the end of the text book.

Term Definition Illustration

Circle Use a ruler and compass to draw a circle with radius 3cm. Label the centre of the circle “O”.

Find the length of the diameter of this circle:_____

Radius

Diameter

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Tangent Line

Draw a tangent line to the circle.

Point of Tangency Identify a point of tangency in the following diagram.

Perpendicular Use a ruler and protractor to draw a pair of perpendicular lines.

Chord Construct 3 chords in this circle. Make one of the chords a diameter.

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A

B

C

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●

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Bisect Construct a chord on this circle. Construct and label 2 bisectors of this chord.

Perpendicular Bisector

Construct and label chord AB on the circle. Construct and label its perpendicular bisector CD.

Label the midpoint of your chord “M”.

Midpoint

Arc

Draw and label arc AC on this circle.

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Draw two angles subtended by this arc.

Subtended

Inscribed Angle

Identify the inscribed angle:_____

Identify the central angle:_____

If the Central angle measures 110°, what is the measure of the inscribed angle?

Central Angle

Semicircle

Draw an angle inscribed in a semicircle.

What is the measure of this angle?

Thanks to Sarah Mathews for sharing this activity.

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Grade 9 Math – Circle Properties

Discovery

We will now use GeoGebra to discover the following circle properties:

1. Tangent-Radius Property2. Perpendicular to Chord Properties3. Central Angle and Inscribed Angle Property4. Inscribed Angle Property5. Angles in a semicircle property

1. Tangent-Radius Property

Using GeoGebra, construct a circle by selecting the following button:

Plot a point outside the circle. Use the following button:

Next, click the following button and select “Tangents”

Click on the point and then the circle. Two tangent lines should appear. Next, we need to find the point of tangency. Select the following button and choose “Intersection of Two Objects”.

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Click on one of the tangent lines and the circle. The intersection point will appear. Next we must draw in the radius between the center of the circle and the point of tangency. Select the following button and choose “segment between two points”.

Your construction should look similar to the following:

Finally, measure the angle between the radius and the tangent line by selecting the following:

Choose three points: the center of the circle, the point of tangency and another point on the tangent line.

What was the measure of the angle? Repeat this process three more times saving your file each time. What did you notice about the angle between the radius and the tangent line? Make a general rule regarding the radius of a circle and a tangent to the circle:

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Tangent – Radius Property

To complete on loose leaf: p. 388 & 389 #'s 3 to 9

2. Perpendicular to Chord Properties

Part 1:

Use the instructions from the previous section to help you construct a circle. Place two points on the circle’s circumference. Form a chord by drawing a line segment between the two points.

Next, select the following button and choose “perpendicular line”:

Your construction should look similar to the following:

Select the centre of the circle and the chord. Next, measure each section of the divided chord by selecting the following button and choosing “distance or length”:

Click on one endpoint of the line segment and then the point where the perpendicular meets the chord. Then, measure the distance between the other endpoint and the intersection of chord and the perpendicular.

What do you notice about the lengths of the two segments of the chord? Repeat this process three more times saving your file each time Make a general rule about dropping a perpendicular line from the center of a circle to a chord.

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Perpendicular to Chord Property 1

Part 2:

Construct another circle. Plot two points on the circle’s circumference and construct a chord. This time, find the midpoint of the chord by selecting the following button and choosing “midpoint”:

Your construction should look similar to the following:

Once you have the midpoint, draw a line segment between the center and the midpoint. Next measure the angle between the center of the circle, the midpoint and one point on the circle.

What do you notice about angle? Repeat this process three more times saving your file each time. Make a general rule regarding the angle formed by drawing a segment from the center of a circle to the midpoint of a chord.

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Perpendicular to Chord Property 2

To complete on loose leaf: p. 397 & 398 #'s 1 to 7, 10

3. Central Angle and Inscribed Angle Property & 4. Inscribed Angle Property

Construct a circle. Choose a minor arc by selecting two points on the circle. Draw line segments between each point and the center. Measure the central angle. Now choose another point on the circumference outside the minor arc. Draw segments from the endpoints of the arc to the third point. This is the inscribed angle.

Your construction should look similar to the following:

What do you notice about the relationship between the inscribed and central angles. Move the third point around on the circumference of the circle. What do you notice about the measure of the inscribed angle? Repeat this process three more times saving your file each time. Make a general rule regarding the relationship between the central angle and the inscribed angle AND make a general rule regarding all inscribed angles subtended by the same arc.

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Central Angle and Inscribed Angle Property

Inscribed Angle Property

5. Angles in a semicircle property

Select the following button and choose “semicircle through two points”:

Connect the two points of the semicircle. This is your diameter and central angle. Choose and plot any point on the circumference of the semicircle. Make line segment between each endpoint of the semicircle and the chosen point. This is your inscribed angle. Measure this angle.

Your construction should look similar to the following:

Move the third point around on the circumference of the semicircle. What do you notice about the measure of the inscribed angle? Repeat this process three more times saving your file each time. Make a general rule about inscribed angles subtended by a semicircle.

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Angles in a Semicircle Property

To complete on loose leaf: p. 410, 411 #'s 3 to 6, 11

Extend your Knowledge

Use what you have learned to solve these problems:

1. Point O is the centre of the circle. Point P is a point of tangency. Determine the values of x and y. Justify your solutions.

2. Point O is the centre of the circle. Determine the values of a and b. Justify your solutions.

3. Point O is the centre of the circle and BE is a diameter.

Determine the values of e, f, and g. Justify your solutions.

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Reflecting on This Activity

1. Read each of the Learning Goals for this activity. Think about your learning. Rate your learning for each goal using these symbols:

+ = I can understand and do this very well. = I can understand and do this all right.– = I am still having some trouble with this. = This is a big problem for me.

2. In this activity, what was most challenging for you or your class? What was difficult about it?

________________________________________________________________________

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What helped you learn the challenging material?

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What could you have done differently that might have made it easier for you?

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3. What one important piece of advice would you give to other students about working with circle geometry?

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Taken From Pearson Education Canada 2009

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Learning Goal My Rating

A tangent to a circle is perpendicular to the radius.

The perpendicular from the centre of the circle to a chord bisects the chord.

The measure of a central angle is twice the measure of inscribed angles subtended by the same arc.