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2.9: Circles

Difficulty Level: At Grade Created by: CK-12
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Parts of Circles & Tangent Lines

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 that distance. 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.

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.

Tangency - Initially students get very confused by the different tangencies. There is a lot here that they need to digest. Keep reviewing the differences between internally and externally tangent circles and tangent lines that are internal and external so that students have a chance to practice identifying the differences.

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.

Using the Pythagorean Theorem to Find Side Measurers in Right Triangles - Using the Pythagorean Theorem to find the measures of sides in a right triangle is a common practice in this section. Students should be on the lookout for right triangles formed by a radius and tangent segment. Later in this chapter they will also find right triangles formed by radii (and diameters) and chords that they bisect. Recognizing these perpendicular segments and the right triangles they form will help students solve problems.

Properties of Arcs

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.

Sum is \begin{align*}360^\circ\end{align*} - Remind students that the entire arc (all the way around the circle) is \begin{align*}360^\circ\end{align*}. This may seem obvious to some students, but others don’t pick up on this right away and need reminding.

Central Angles - Students may grasp the idea immediately that central angles are equal in measure to the arc they intercept, but they often forget this property as they add additional angle theorems for circles. There are so many different angles that can be formed in circles and this is just the first to be explored. It is imperative that students focus on the fact that central angles have a vertex at the center of the circle (not on the circle or in the circle). Keep saying this every time you talk about a central angle so that students really internalize this definition. If they continually go back to where the vertex is located they will be better able to distinguish between the different types of angles formed by segments in, on and around a circle.

Properties of Chords

Update the Theorem List - Students should be keeping a notebook full of all the theorems they have learned in geometry class. The outlines of keywords, definitions and theorems provided at the beginning go each chapter is an excellent basis for this. 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.

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 to get the arc measure (or divide the arc measure by 2 to get 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 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 \begin{align*}180^\circ\end{align*}.
  2. A convex quadrilateral with a sum of \begin{align*}360^\circ\end{align*}.
  3. A right triangle formed with a tangent and radius
  4. An isosceles triangle formed with two radii
  5. A diameter creating a semicircle
  6. Arcs covering the entire circle
  7. Central or Inscribed angles
  8. Congruent tangents
  9. A right angle inscribed in a semicircle
  10. Perpendicular segments (radius and tangent, radius and chord that is bisected)
  11. Similar triangle with proportional sides
  12. 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 be 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. After that, lots of practice is advised. This unit is often very difficult for even the strongest math students simply because of the amount of information they must absorb and apply.

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 \begin{align*}360^\circ\end{align*}-(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.

Example 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: \begin{align*}60^\circ\end{align*}.

Make a sketch to illustrate the problem as shown here.

\begin{align*}x+2x&=360\\ x&=120\\ \text{angle measure}& = (240-120) \div 2=60^\circ\end{align*}

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

Answer: Central angles. The chords must intersect at the center of the circle in order to intercept congruent arcs, which makes the angles formed 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.

Go Through the Proportions - Take the time, if you can, to go through the process of proving these relationships between the lengths of secants, chords and tangents using similar triangles. It is not necessary for students to memorize these proofs, but sometimes they have an easier time remembering the properties if they understand their origins.

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.

Example 1: A secant and a tangent segment have a common exterior endpoint. The secant has a total length of 12 cm and the tangent has length 7 cm. What is the measure of both segments of the secant?

Answer: The secant is composed of two segments with approximate lengths 4.1 cm and 7.9 cm.

Let one segment of the secant be \begin{align*}x\end{align*}, so the other can be represented by \begin{align*}20-x\end{align*}.

\begin{align*}7^2&=(12-x) \times 12\\ x & \approx 7.9\end{align*}

Example 2: Two secant segments have a common endpoint outside of a circle. One has interior and exterior segments of lengths 10 ft and 12 ft respectively and the other has a total measure of18 ft. What is the measure of the two segments composing the other secant?

Answer: The secant is composed of two segments with lengths \begin{align*}3 \frac{1}{3} \ ft\end{align*} and \begin{align*}14 \ \frac{2}{3}\ ft\end{align*}.

\begin{align*}12(10+12)&=(18-x) \times 18\\ x&=3 \frac{1}{3}\end{align*}

Practice, Practice, Practice - Perhaps the most effective method to get students to internalize the concepts in this chapter is to provide extensive practice. There is just so much information that students are going to have a hard time remembering it all unless they practice it repeatedly. Mixed up the different concepts in a practice assignment so that they get in the habit of switching gears repeatedly and seeing different types of problems on the same page. Too often, students will just assume that all the problems in a particular assignment should be solve the same way and they don’t stop to think about how each problem might be different from the previous one. Mixing problems up will help them practice this important skill.

Extension: Writing and Graphing the Equations of Circles

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.

Remembering the Equation - Refer back to the definition of a circle- set of all point in the plane equidistant from a given point. This implies that the equation requires this “given” point, or center, and the distance, or radius, of the circle. Show students that the formula is derived by taking a point on the circle, \begin{align*}(x,y)\end{align*}, the center, \begin{align*}(h,k)\end{align*}, and setting the distance between them equal to the radius as shown below.

\begin{align*}\sqrt{(x-h)^2+(y-k)^2 }=r\end{align*}

Now, square both sides to get the equation: \begin{align*}(x-h)^2+(y-k)^2=r^2\end{align*}

The more students know and understand about the origins of particular equations and properties, the more likely they are to remember them in the future. Going through this process also helps students realize that all these equations are not just “made up”, they have roots in concepts they may already know and understand.

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.

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