2.9 Circles
About Circles
Circle Vocabulary –This section has quite a few vocabulary words. Some the students will already know, like
radius, and some, like secant, will be new. Encourage the students to make flashcards or a vocabulary list. They
should know the word definition and have pictures drawn and labeled. It is also important for students to know the
relationships between the words. The radius is half the length of the diameter and the diameter is the longest chard
in a circle. Make knowing the vocabulary a specific assignment, otherwise many students will forget to take the time
to learn the vocabulary well.
Circle or Disk –The phrase “a point on the circle” is commonly used. This will confuses the students that do not
realize that the circle is the set of pointsexactlysome set distance from the center, and not the points less than or
equal to that distance from the center or the circle. What is happening is that they are confusing the definition of
a disk and a circle. Emphasize to the students that a circle is one dimensional; it only contains the points on the
edge. Another option is to give them the definition of a disc along with that of a circle, so that they can compare and
contrast the two definitions.
Inscribed or Circumscribed –An inscribed circle can also be described as a circumscribed polygon. The different
ways that these vocabulary words can be used can make learning the relationships complicated. As a guide, tell the
students that the object inscribed is on the inside. Starting with that, they can work out the rest. For practice, ask the
students to draw different figures that are described in words, like a circumscribed hexagon, or a circle inscribed in
an octagon.
Square the Radius –When working with the equation of a circle, students frequently forget that the radius is
squared in the equation, especially when the radius is an irrational number. Explaining the equation of the circle in
terms of the Pythagorean theorem will help the students remember and understand how to graph this conic section.
Completing the Square –Completing the square to put the equation of a conic section in standard form is a nice
little math trick. It exemplifies the kinds of moves mathematicians use to manipulate expressions and equations.
Students find it difficult to do especially when fractions are involved and they have trouble retaining the process for
more than a few days. Give them many opportunities to practice.
Tangent Lines
Bringing It All Together –This section makes use of many concepts students have previously learned in the class.
It will help students to start to prepare for the final, or for an end of the quarter cumulative test. Students will need
time to go back and review the topics used in this section as well as the normal time allotted to learn the new material.
Below is a list of subjects the students must be competent at to be successful with this section. A day spent reviewing
these will help avoid frustration.
Review Topics:
a. The converse of a conditional statement and proof by contradiction
b. Proofs that employ congruent triangles and there corresponding parts
c. The Pythagorean theorem and its converse2.9. Circles