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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 points exactly some 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:

  1. The converse of a conditional statement and proof by contradiction
  2. Proofs that employ congruent triangles and there corresponding parts
  3. The Pythagorean theorem and its converse
  4. Equations of lines and circles, including slopes of perpendicular lines
  5. The proportionality of the sides of similar triangles
  6. Polygons: the sum of interior angles and regular polygons

All the Radii of a Circle Are Congruent – It may seem obvious, but frequently students forget to use the fact that all the radii of a circle are congruent. This follows directly from the definition of a circle. Remind students to use this fact when setting up equations and assigning variables to different radii in the same circle.

Congruent Tangents In this section the Tangent Segment Theorems is proved and applied. Remind student that this is only true for tangents and does not extend to secants. Sometimes student will see a secant enter a circle and think the distance from the exterior point to where the secant intersects the circle is the same as a tangent or another secant from that same point.

Hidden Tangent Segments – Sometimes it is difficult for students to recognize tangent segments because they are imbedded in a more complex figure, or the tangent segment is extended in some way. A common situation where this occurs is when there is an inscribed circle. Tell the students to be on the lookout for tangent segments. They should look at segments individually and as part of the whole. Sometimes it is helpful to use a small sticky note to cover parts of the figure so they do not distract from the area of focus.

Common Tangents and Tangent Circles

Using Trigonometry to Find Side Measurers in Right Triangles – Using the definitions of sine, cosine, or tangent to find the measures of sides in a right triangle is a common application of trigonometry that is put to use in this section. Students will need a bit of practice and perhaps a step-by-step process when learning this skill. With some experience though, this will become an easy, enjoyable task.

Step-by-Step Process:

  1. Highlight the side of the right triangle that’s measure is to be found. Place a variable, say x, by that side.
  2. Chose one of the acute angles of the right triangle whose measure is known to work from. Highlight this angle in another color.
  3. Chose another sides of the triangle whose measure is known. Highlight that side in the same color as the other side.
  4. Decide what relationships (opposite leg, adjacent leg, or hypotenuse) the highlighted sides have to the highlighted angle.
  5. Decide which of the three trigonometric ratios utilize those side relationships.
  6. Write out the definition of that trigonometric ratio.
  7. Substitute in the highlighted values.
  8. Solve the equation by either multiplying or dividing. It is best to not round the decimal approximation of the trigonometric ratio taken from the calculator. Round after the multiplication or division has taken place.

Key Exercises:

1. \triangle {ABC} is a right triangle with the right angle at vertex C.

m \angle A=52^\circ and AC = 10 \;\mathrm{cm}. Find BC.

Answer:

\text{tan}\ 52^\circ =\frac{BC}{10} && BC \approx 12.8\ \text{cm}

2. \triangle {DEF} is a right triangle with the right angle at vertex F.

m \angle D=22^\circ and DF = 1143 \;\mathrm{ft}. Find DE.

Answer:

\text{cos}\ 22^\circ =\frac{1143}{DE} && DE \approx 1200\ \text{ft}

Arc Measure

Naming Major Arcs and Semicircles – When naming and reading the names of major arcs and semicircles, the three letter system is sometimes confusing for students. When naming an angle with three letters, the first place to look is to the middle letter, the vertex. It is just the opposite for a three letter arc name. First, the students should locate the endpoints of the arc at the ends of the name. For a major arc they have two arcs to choose from. The major arc uses three letters and is the long way around. Any of the other points on the major arc can be used to designate that the long path is being taken. A semicircle divides the circle into two congruent arcs. A third letter is needed to designate which half of the circle is being named.

Look For Diameters – When working exercises that call for students to find the measures of arcs by adding and subtracting arc and angle measures in a circle, students often forget that a diameter divides the circle in half, or into two 180 degree arcs. Remind the students to be on the lookout for diameters when finding arc measures.

Using Trigonometry to Find Angle Measures in Right Triangles – A similar process is needed for finding angles in right triangles as for finding sides in right triangles in the previous lesson. Students need some scaffolding when they first learn to use this method.

  1. Highlight the angle whose measure is to be found.
  2. Two sides of the right triangle must be known. Highlight these two sides.
  3. Decide what relationship the highlighted sides have to the angle in question.
  4. Decide which trigonometric ratio used those side relationships.
  5. Write and solve an equation. Remember to use the inverse of the trigonometric ratio on the calculator since it is the angle that needs to be found.

Key Example:

1. \triangle {ABC} is a right triangle with the right angle at vertex C.

AC = 12 \;\mathrm{cm}, and AB = 17 \;\mathrm{cm}. Find m\angle B.

Answer:

\text{sin}\ B=\frac{12}{17} && m \angle B \approx 45^\circ

Chords

Update the Theorem List – Students should be keeping a notebook full of all the theorems they have learned in geometry class. These theorems are like tools that can be used to work exercises and write proofs. This section has quite a few different theorems about the relationships or chords and angles that need to be included in their notebook. Each entry should have the name of the theorem, the written statement of the theorem, and a picture to illustrate the 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.

  1. 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 the y-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.
  2. In the same example the quadratic formula is used to solve for the two possible values of the x-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.

  1. Draw in segments to create right triangles, central angles, and any other useful geometric objects.
  2. 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.
  3. Use trigonometry of right triangles to find the angles and segment lengths needed to complete the exercise.
  4. 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.
  5. 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?

  1. A triangle with a sum of 180 \;\mathrm{degrees}
  2. A convex quadrilateral with a sum of 360  \;\mathrm{degrees}
  3. A right triangle formed with a tangent
  4. An isosceles triangle formed with two radii
  5. A diameter creating a 180  \;\mathrm{degree} semicircle
  6. Arcs covering the entire 360  \;\mathrm{degrees} of the circle
  7. Central or Inscribed angles
  8. Tangents that form right angles
  9. Similar triangle with proportional sides
  10. Congruent triangles with congruent corresponding parts

Any New Information is Good – If students can not immediately see how to find the measure they are after, advise them to find any measure they can. This keeps their mind active and working. Frequently, they will be able to use the new information to find other measures, and will eventually work their way around to the desired answer. This might not br the most efficient method, but the students’ technique will improve with practice.

Angles of Chords, Secants, and Tangents

Where’s the Vertex? – When determining the relationships between angles and arcs in a circle the location of the vertex of the angle is the determining factor. There are four possibilities.

  1. The vertex of the angle is at the center of the circle, it is a central angle, and the arc and angle have the same measure.
  2. The vertex of the angle is on the circle. The angle could be made by two cords, an inscribed angle, or by a chord and a tangent. In either situation, the measure of the arc is twice that of the angle.
  3. The vertex of the angle is inside the circle, but not at the center. In this case two arcs are necessary, and the angle measure is the average of the measures of the arcs cut off by the chords that form the vertical angles.
  4. The vertex of the angle is outside the circle. Then the two intersected arcs have to be subtracted and the difference divided by two. Note the similarity to an average.

Students often need help organizing information in this way. It is best to do this with them, as a class activity so that in the future they will be able to do it for themselves.

Use the Arcs – It is typical to have more than one angle intercepting a specific arc. In this case a measure can be moved to an arc and then back out to another angle. Another situation students should look for is when a circle is divided into two arcs. One arc can be represented as 360 - (an expression for the other arc). Students sometimes miss these kinds of moves. It may be beneficial to have students share with the class the different strategies and patterns they see when working on these exercises.

Additional Exercises:

1. Two tangent segments with a common endpoint intercept a circle dividing it into two arcs, one of which is twice as big as the other. What is the measure of the angle formed by the by the two tangents?

Answer:

x + 2x & = 360 && \text{angle measure} = (240 - 120) \div 2 = 60\ \text{degrees} \\x &= 120

2. Two intersecting chords intercept congruent arcs. What kind of angles do the chords form?

Answer: central angles

Segments of Chords, Secants, and Tangents

Chapter Study Sheet – This chapter contains many relationships for students to remember. It would be helpful for them to summarize all of these relationships on a single sheet of paper to use when studying. Some instructors allow students to use these sheets on the exam in order to encourage students to make the sheets. The value of a study sheet is in its making. Students should know this and make them regardless of whether they can be used on the exams. Sometimes if students know that they will be able to use the study sheet, they will not work to remember all of the relationships, and their ability to learn the material is compromised. It is a hard issue to work around and each instructor needs to deal with it as he or she feels best with their particular classes.

When to Add – When writing proportions involving secants, students will have a difficult time remembering to add the two segments together to form the second factor. A careful study of the proof will help them remember this detail. When they see secants, have them picture the similar triangles that could be drawn. Remind them, and give them ample opportunity to practice.

Have Them Subtract – One way to give students more practice with the lengths of secants in circles is to give them exercises where the entire length of the secant is given, and they have to setup an expression using subtraction to use in the proportion.

Key Examples:

1. A secant and a tangent segment have a common exterior endpoint. The secant has a total length of 12 \;\mathrm{cm} and the tangent has length 7 \;\mathrm{cm}. What is the measure of the both segments of the secant?

Answer:

Let one segment of the secant be x, so the other can be represented by 20 - x.

7^2 & = (12 - x)*12 && \text{The secant is composed of two segments}\\&&& \text{with approximate lengths of}\ 4.1 \ \text{cm and}\ 7.9 \ \text{cm}\\x & \approx 7.9

2. Two secant segments have a common endpoint outside of a circle. One has interior and exterior segments of lengths 10 \;\mathrm{ft} and 12 \;\mathrm{ft} respectively and the other has a total measure of 18 \;\mathrm{ft}. What is the measure of the two segments composing the other secant?

Answer:

12(10 + 12) & = (18 - x)*18 && \text{The secant is composed of two segments}\\&&& \text{with lengths}\ 3\frac{1}{3} \ \text{ft and}\ 14\frac{2}{3}\ \text{ft}\\x & = 3\ \frac{1}{3}

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