Lesson 5 Circular Reasoning Practice Understanding

Learning Focus

Apply circle geometry theorems in various contexts.

How do I examine a complex geometric figure for the structures and features that will support my reasoning?

How do I use the properties and theorems of circle geometry to model real-world contexts?

Open Up the Math: Launch, Explore, Discuss

The following problems will draw upon your knowledge of similarity, circle relationships, and trigonometry.

1.

In the diagram, the radius of $⨀ D$ is $5$ cm and $F$ is the midpoint of $\overset{―}{A E}$ . Segments $\overset{―}{G E}$ and $\overset{―}{G A}$ are tangent to $⨀ D$ . The measures of arc $E B$ and arc $B C$ are given in the diagram. Find the measures of all other unmarked angles, arcs, and segments.

Circle Modeling Problems

2.

Now you see it...

A lot of circular geometry is embedded within the human eye. Light rays reflected off of objects in our surroundings enter the eye through the cornea and are bent by the lens of the eye to focus them on the retina at the back of the eye. Use the following information to find the inscribed angle in this simplified diagram of the adult human eye.

The circumference of the eye is $7.32 cm$ .
The cornea measures $1.22 cm$ .

3.

… and now you don’t.

Analyzing planetary motion is an important part of mathematics. For example, predicting the timing of an eclipse can involve some circular geometry.

Based on their everyday experience, ancient astronomers believed that the sun traveled around the earth in a circular orbit, just as the moon does. While Galileo and Copernicus discovered evidence to change this model in the 1600’s, the Earth-centered model of the sun’s orbit was useful for measuring the distance to the sun and the moon in ancient times, as well as for calculating their diameters and determining their positions relative to each other. We will use their modeling perspective in the following problem.

A solar eclipse is a surprising result of the fact that although the diameter of the sun is approximately $400$ times bigger than the diameter of the moon, the sun is also approximately $400$ times farther away from the earth than the moon. Consequently, when the path of the moon crosses in front of the sun, the moon completely hides the face of the sun. As shown in the diagram, the intercepted arc of the moon’s orbit that is occupied by the moon is the same as the intercepted arc of the sun’s apparent orbit that is occupied by the sun. The measure of this intercepted arc for a person on the surface of the earth is approximately $0.5 °$ , and the diameter of the moon is about $2,100$ miles.

Using the information given above, along with the following diagram of the relative positions of the earth, moon, and sun during a total eclipse, approximate the distance from the earth to the moon and the earth to the sun.

Ready for More?

In the diagram provided, $△ A B C$ is equilateral. All circles are tangent to each other and to the sides of the equilateral triangle. The radius of the three smaller circles, $⨀ P$ , $⨀ Q$ , and $⨀ R$ , is $4 cm$ . The radius of $⨀ O$ is not given.

Find the radius of $⨀ O$ and the length of the sides of the equilateral triangle.

Hint: There are a lot of equilateral triangles in this diagram. An altitude of an equilateral triangle bisects an angle and the opposite side. How would the side lengths of this $30 ° - 60 ° - 90 °$ triangle relate to the side lengths of the equilateral triangle from which it was formed?

Takeaways

My procedure for finding the length of the tangent segments drawn from the vertex of the circumscribed angle to the points of tangency:

Two different measurements are associated with an arc: the measure of the arc and the arc length.

The measure of the arc is found by , and the unit of measure is .

The arc length is found by , and the unit of measure is .

Vocabulary

arc length
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we applied theorems about inscribed and circumscribed angles of a circle to find the lengths of many segments associated with the circle, including the lengths of the tangent segments drawn from the vertex of a circumscribed angle to the points of tangency on the circle. To find these lengths, we often had to draw upon right triangle trigonometry.

Retrieval

1.

Determine the angles of rotational symmetry and the number of lines of reflective symmetry for the regular heptagon.

2.

Find the missing side lengths and angle measurements.

3.

Segment $A C$ is the diameter of circle $B$ . $A C = 30 ft$ .

a.

Find the area and the circumference of circle $B$ .

b.

Find the distance to walk along arc $A C$ .

c.

Find the distance to walk from $A$ to $B$ to $C$ and then from $C$ around the outside of the circle back to $A$ .

d.

Find the area of half of the circle.