What if you were asked to geometrically consider the ancient astronomical clock in Prague, pictured below? It has a large background circle that tells the local time and the “ancient time” and then the smaller circle rotates around on the orange line to show the current astrological sign. The yellow point is the center of the larger clock. How does the orange line relate to the small and larger circle? How does the hand with the moon on it (black hand with the circle) relate to both circles? Are the circles concentric or tangent? After completing this Concept, you'll be able to use your knowledge of parts of circles to answer questions like these.
For more information on this clock, see: http://en.wikipedia.org/wiki/Prague_Astronomical_Clock
Watch the first half of this video.
A circle is the set of all points in the plane that are the same distance away from a specific point, called the center. The center of the circle below is point . We call this circle “circle ,” and it is labeled .
Important Circle Parts
Radius: The distance from the center of the circle to its outer rim.
Chord: A line segment whose endpoints are on a circle.
Diameter: A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.
Secant: A line that intersects a circle in two points.
Tangent: A line that intersects a circle in exactly one point.
Point of Tangency: The point where a tangent line touches the circle.
The tangent ray and tangent segment are also called tangents.
Tangent Circles: Two or more circles that intersect at one point.
Two circles can be tangent to each other in two different ways, either internally tangent or externally tangent.
If the circles are not tangent, they can share a tangent line, called a common tangent. Common tangents can be internally tangent and externally tangent too. Notice that the common internal tangent passes through the space between the two circles. Common external tangents stay on the top or bottom of both circles.
Concentric Circles: Two or more circles that have the same center, but different radii.
Congruent Circles: Two or more circles with the same radius, but different centers.
Find the parts of that best fit each description.
a) A radius
b) A chord
c) A tangent line
d) A point of tangency
e) A diameter
f) A secant
b) , or
Draw an example of how two circles can intersect with no, one and two points of intersection. You will make three separate drawings.
Determine if any of the following circles are congruent.
From each center, count the units to the outer rim of the circle. It is easiest to count vertically or horizontally. Doing this, we have:
From these measurements, we see that .
Notice the circles are congruent. The lengths of the radii are equal.
Watch this for help with the Examples above.
Concept Problem Revisited
Refer to the photograph in the “Concept” section at the beginning of this Concept. The orange line (which is normally black, but outlined for the purpose of this exercise) is a diameter of the smaller circle. Since this line passes through the center of the larger circle (yellow point, also outlined), it is part of one of its diameters. The “moon” hand is a diameter of the larger circle, but a secant of the smaller circle. The circles are not concentric because they do not have the same center and are not tangent because the sides of the circles do not touch.
A circle is the set of all points that are the same distance away from a specific point, called the center. A radius is the distance from the center to the outer rim of a circle. A chord is a line segment whose endpoints are on a circle. A diameter is a chord that passes through the center of the circle. The length of a diameter is two times the length of a radius. A secant is a line that intersects a circle in two points. A tangent is a line that intersects a circle in exactly one point. The point of tangency is the point where the tangent line touches the circle. Tangent circles are two or more circles that intersect at one point. Concentric circles are two or more circles that have the same center, but different radii. Congruent circles are two or more circles with the same radius, but different centers.
1. If the diameter of a circle is 10 inches, how long is the radius?
2. Is it possible to have a line that intersects a circle three times? If so, draw one. If not, explain.
3. Are all circles similar?
1. The radius is always half the length of the diameter, so it is 5 inches.
2. It is not possible. By definition, all lines are straight. The maximum number of times a line can intersect a circle is twice.
3. Yes. All circles are the same shape, but not necessarily the same size, so they are similar.
Determine which term best describes each of the following parts of .
- What is the longest chord in any circle?
Use the graph below to answer the following questions.
- Find the radius of each circle.
- Are any circles congruent? How do you know?
- How are and related?
- Find the equation of .
- Find the length of .