Name
Ti-nspire Activity – Circles Exploration
Date:
Period:
Open the TI-Nspire document Circles_Angles_and_Arcs.tns.
A circle is the set of all points in a plane that are equidistant from a
given point in the plane. Circles, angles, and arcs have many
interesting characteristics. In this activity, you will explore relationships
among different types of angles and arcs in a circle.
Press / ¢ and / ¡ to
Move to page 1.2.
navigate through the lesson.
1. Drag point A or point C. Describe the changes that occur in the figure as you drag the point.
2. Angle AOC is an example of a central angle. What makes it a central angle?
An angle intercepts an arc of a circle if each endpoint of the arc is on a different ray of the angle and the
other points of the arc are in the interior of the angle.
Move to page 1.3.
As you move point A or point C, the central angle AOC intercepts a minor arc AC. The measure of the
minor arc equals the measure of the central angle. The larger remaining arc, ABC, is called a major arc.
3. a. Move point A or point C to help you complete the table.
AOC
arc AC
50
50
arc ABC
arc AB + arc ABC
100
250
(Choose an angle.)
b. What is true about the AB + ABC, the sum of the measures of the minor and major arcs?
4. In a circle, the measure of a central angle AOC is n.
a. What is the measure of the minor arc that is intercepted by the central angle? How do you know?
b. What is the measure of the major arc? How do you know?
Move to page 1.4.
5. Angle ABC is called an inscribed angle because BA and BC are chords of the circle and vertex B is
on the circle. Drag point B around the circle.
a. As point B is moved around the circle, what do you notice about the measure of ABC?
b.
Why does mABC change when point B is moved from one arc to the other? Explain your
reasoning.
c.
Move point A or point C until ABC is a right angle. What is special about the arc and AC ?
Move to page 1.5.
Angle ABC intercepts arc AC. Drag point D to various locations outside the circle, on the circle, inside the
circle, and at the center O.
6. Place point D on the circle so that ADC intercepts the same arc as ABC.
a. What do you notice about the measures of ABC and ADC?
b. What happens to the angles if you move point A or point C?
7. Place point D at the center of the circle. Move point A and point C so that ADC intercepts the same
arc as ABC.
a. What is the relationship between the measures of inscribed ABC and central ADC?
8. Place point D on the circle so that ABCD is a quadrilateral.
a. What do you notice about the sum of the measures of ABC and ADC? Check with a classmate
to compare.
b. What do you notice about the sum of the measures of the angles if you move point A or point C?
c.
What do you notice about arcs ABC and ADC?
d. How does the relationship between arcs ABC and ADC explain the sum of the measures of
inscribed ABC and ADC?
Open the TI-Nspire document Application_of_a_Circle_Angles_
and_Arcs.
In this activity, you will solve a real-world application involving a
circle. This will require using the relationships between central
angles, inscribed angles, and the arcs they intercept.
Move to page 1.2.
Suppose you work for an architectural firm and a new business complex is in the process of
being designed. The plans for the complex include a circular courtyard within a square area
with side lengths of 8.4 yards. The courtyard will use 10-inch square pavers in different
colors to create a design, as shown in the diagram on page 1.4. The points A, B, C, D, and
E represent the points of the star design, and each of the points lies on the circle. Point Q
represents the center of the circle.
You have been asked to supply the company constructing the courtyard some information
that will help with creating the design and ordering supplies.
Move to page 1.4.
1. Would the angles A, B, C, D, and E be considered central angles or inscribed angles?
Explain.
2. What is the relationship between the measures of the angles A, B, C, D, and E and the
arcs they intercept? Explain.
3. Would the arcs intercepted by each of the angles A, B, C, D, and E be considered
major arcs or minor arcs? Explain.
4. In order for the star pattern to be uniform, each of the angles should have the same
degree measure. What should be the degree measure of each of the angles A, B, C, D,
and E? Explain your reasoning.
5. Grab and move points A, B, C, D, and E around the circle until all angle measures are
the same. What is the degree measure? Is this what you expected? Explain.
6. What is the diameter of circle Q? Explain your reasoning.
Check your answer to Question 6 by using the Segment tool (MENU > Points & Lines > Segment) to
create a segment that represents the diameter of circle Q. Use the Length tool (MENU > Measurement
> Length) to find the length of the segment. Change the Attributes (MENU > Actions > Attributes) of
the measurement to one decimal place.
7. Given that the circumference of a circle can be found by using the formula C = 2 r, where r is the
radius, find the circumference of circle Q to the nearest yard. Show your work below.
Check your answer to Question 7 by using the Length tool to find the length of circle Q. Change the
Attributes of the measurement to display zero decimals.
8. Given that the area of a circle can be found by using the formula A =  r2, where r is the radius, find
the area of circle Q to the nearest yard. Show your work below.
Check your answer to Question 8 by using the Area tool (MENU > Measurement > Area) to find the
area of circle Q. Change the Attributes of the measurement to display zero decimals.
9. Given that the pavers being used to construct the courtyard are squares with side lengths of 10
inches, how many pavers will need to be ordered to construct the courtyard? (1 yd2 = 1,296 in.2)