relationship. Not only will this be good reference material, making the notebook will help the students to remember
the material.
Algebra Review –Students may need a bit of a review before correctly squaring algebraic expressions and solving
quadratic equations in geometric applications.
a. In example three the equation of a line is substituted into the equation of a circle so points of intersection
can be found. When the binomial is substituted for they−variable in the circle equation, it must be squared.
Students frequently try to “distribute” the square instead of using the FOIL method. Make a point of writing
out the binomial twice, and multiplying. Students should know and be able to use the pattern for a perfect
square binomial, but they will understand why they have to use the pattern when they see the long way written
out once and awhile, and will be more likely to remember.
b. In the same example the quadratic formula is used to solve for the two possible values of thex−variable.
Students will benefit from a brief explanation of how quadratic equations are solved. First, when the student
realized that it is a second degree equation they need to solve for zero. Then the equation can be factored or
the quadratic formula can be applied. The students should remember the process quickly when they see it.
This is an important topic of algebra, and it is always good to review to eliminate misconceptions.Tips and Suggestions –There are a few strategies that students should keep in mind when working on the exercises
in this section.
a. Draw in segments to create right triangles, central angles, and any other useful geometric objects.
b. Remember to split the length of the chord in half if only half of it is used in a right triangle. Don’t just use
the numbers that are given. The theorems must be applied to get the correct number, and multiple steps will
usually be necessary.
c. Use trigonometry of right triangles to find the angles and segment lengths needed to complete the exercise.
d. Don’t forget that all radii are congruent. If you have the length of one radius, you have them all, including the
ones you add to the figure.
e. Employ the Pythagorean theorem and any other tool you have from previous lessons that might be useful.Inscribed Angles
Inscribed Angle or Central Angle –When students spot an arc/angle pair to use in solving a complex circle
exercise, the first step is to identify the angle as a central angle, an inscribed angle, or possibly neither. If necessary,
they can trace the sides of the angle back from the arc to see where the vertex is located. If the vertex is at the center
of the circle, it is a central angle, and the measure of the arc and the angle are equal. If the vertex is on the circle, it
is an inscribed angle, and the students must remember to double the angle measure. A good mnemonic device is to
think of the arc of an inscribed angle being farther away from the vertex than the arc of a central angle. Therefore the
measure of the arc will be larger. If the vertex is at neither the exact center or on the circle, no arc/angle relationship
can be determined with only one arc.
What to Look For –Students can be overwhelmed by the number of different relationships that need to be used to
solve these circle exercises. Sometimes they can just get paralyzed and not know where to start. In small groups, or
as a class, have them create a list of possible tools that are commonly used in these types of situations.
Does the figure contain?
a. A triangle with a sum of 180 degrees
b. A convex quadrilateral with a sum of 360 degrees
c. A right triangle formed with a tangent
d. An isosceles triangle formed with two radiiChapter 2. Geometry TE - Common Errors