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1.9: Circles

Difficulty Level: At Grade Created by: CK-12
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Day 1 Day 2 Day 3 Day 4 Day 5

Parts of Circles & Tangent Lines

Investigation 9-1

Finish Parts of Circles & Tangent Lines

Start Properties of Arcs

Finish Properties of Arcs

Quiz 1

Start Properties of Chords

Finish Properties of Chords

Investigation 9-2

Investigation 9-3

Day 6 Day 7 Day 8 Day 9 Day 10

Inscribed Angles

Investigation 9-4

Finish Inscribed Angles

Investigation 9-5

Quiz 2

Start Angles from Chords, Secants, and Tangents

Investigation 9-6

Finish Angles from Chords, Secants and Tangents

Investigation 9-7

Investigation 9-8

Segments from Secants and Tangents
Day 11 Day 12 Day 13 Day 14 Day 15

Quiz 3

\begin{align*}*\end{align*}Start Extension: Equations of Circles

\begin{align*}*\end{align*}Finish Extension:

Equations of Circles

Review for Chapter 9 Test Chapter 9 Test Start Chapter 10

Parts of Circles & Tangent Lines


This lesson introduces circles and several parts of circles. Then, we will explore the properties of tangent lines.

Notation Note

\begin{align*}\odot\end{align*} is the symbol for a circle. A circle is labeled using this symbol and its center. Some textbooks place a dot in the center, some do not. Do not confuse this symbol with an uppercase \begin{align*}\bigcirc\end{align*}.

Teaching Strategies

There is a lot of vocabulary in this chapter. Encourage students to make flash cards for each word and the theorems. Give students index cards at the end of each class period so they can make flash cards in the last 5-10 minutes of all the terms they learned.

Discuss with students where one could find tangent circles, tangent lines, concentric circles, and intersecting circles in real life. Concentric circles can also be formed when raindrops hit a body of water, such as a lake or puddle. An example of intersecting circles can be a Venn diagram. Common external tangent lines could be the gears of a bicycle.

Investigation 9-1 can be done as a teacher-led activity or individually. For step 2, it can be difficult for students to draw a perfectly tangent line at point \begin{align*}B\end{align*}. You may need to circulate to ensure they each student has the correct drawing, to ensure that they get the correct measurement of \begin{align*}\angle ABC\end{align*}. After measuring \begin{align*}\angle ABC\end{align*}, ask students if they think this will happen at angle drawn to the point of tangency. Feel free to repeat this investigation with a larger or smaller circle so students can see two different cases. Lead them towards the Tangent to a Circle Theorem.

In this lesson and chapter, students will need to apply the Pythagorean Theorem. Having a tangent line and a radius that meet at the same point will always produce a \begin{align*}90^\circ\end{align*} angle. Therefore, students will need to recall all the information they learned in the last chapter (special right triangles, Pythagorean Theorem and its converse, and the trig ratios). Review these points, if needed, and proceed with Example 4.

Example 6 is an important example because it uses the converse of the Pythagorean Theorem to show that \begin{align*}B\end{align*} is not a point of tangency. Make sure students understand this point. In Example 7, students will need to draw an additional line, \begin{align*}\overline{EB}\end{align*}. Encourage students to draw in additional lines if ever needed.

If students wonder the reasoning behind Theorem 9-2, draw in radii \begin{align*}\overline{AD}\end{align*} and \begin{align*}\overline{AB}\end{align*} and segment \begin{align*}\overline{AC}\end{align*}. You can give students this additional example.

Additional Example: Put the reasons for the proof of Theorem 9-2 in the correct order.

Solution: \begin{align*}B, E, D, F, C\end{align*}, and \begin{align*}A\end{align*}.

Properties of Arcs


This lesson introduces arcs, central angles, their properties and how to measure them.

Notation Note

Use a curved line above the endpoints of the arc to label it. Some arcs, such as major arcs, require three letters to label it. Explain to students that they must use three letters for some arcs to distinguish between the two different arcs that have the same endpoints. If only two letters are used to label an arc, it is assumed that the arc is less than \begin{align*}180^\circ\end{align*}.

Teaching Strategies

When introducing arcs, discuss with students where they might see arcs and degrees in real life. Examples are pizza crust, pie crust, a basketball hoop (the rim), among others. An arc could be a piece of pizza crust or pie crust. Some students might be familiar with skateboarding or snowboarding. A \begin{align*}360^\circ\end{align*} jump is one rotation, a \begin{align*}540^\circ\end{align*} is one-and-a-half rotations, and a \begin{align*}720^\circ\end{align*} is two full rotations. The link below is Shaun White from the 2010 Winter X Games. In the second and third jumps, see if students can determine how many rotations he does. http://www.youtube.com/watch?v=phZow-WZ96A

Make sure that students are comfortable with finding arc measures using central angles. This is the beginning of arc measures and corresponding angles in circles, so having a solid foundation is important. In Example 3, there are more congruent arcs than the ones that are listed. See if students can find other possibilities: \begin{align*}\widehat{DAE} \cong \widehat{DBE} \cong \widehat{ADB} \cong \widehat{AEB}\end{align*} (all semicircles), \begin{align*}\widehat{ADE} \cong \widehat{DAB}\end{align*}, and \begin{align*}\widehat{DBA} \cong \widehat{EDB}\end{align*} This would also be a good time to discuss different ways to label the same major arc. For example, \begin{align*}\widehat{ADE}\end{align*} from above could also have been labeled \begin{align*}\widehat{ABE}\end{align*}.

The Arc Addition Postulate should be familiar to students; it is very similar to the Segment Addition Postulate and the Angle Addition Postulate. Rather than adding segments or angles, we are now adding arcs. This is a very useful postulate when trying to find all the arcs in a circle.

Properties of Chords


Students will find the lengths and learn the properties of chords in a circle.

Notation Note

There is no explicit way to mark that two arcs are congruent in a picture. Students will have to infer from other information if two arcs are congruent or not. Ways that they can tell are: if corresponding chords are congruent or if the central angles are congruent.

Teaching Strategies

Make sure students understand all the possible ways to interpret Theorem 9-3. First, if the chords are congruent, then the arcs are congruent. Second, the converse is also true (you may need to review “if and only if” and biconditional statements). Finally, you can also say that if the central angles are congruent, then the arcs are congruent AND the chords are congruent. The converse of this statement is also true.

In Investigation 9-2, the construction tells us that a diameter bisects a chord. From this, students should also see that the diameter is perpendicular. Therefore, the perpendicular bisector of a chord is also a diameter (Theorem 9-4). Stress to students that other chords can be perpendicular to a chord and other chords can bisect a chord, but only the diameter is both. Remind students that every line segment has exactly one perpendicular bisector. Also, students should understand that not every diameter drawn that intersects a given chord will be its perpendicular bisector. This investigation is best done as a teacher-led activity.

Like Theorem 9-3, make sure students understand all the possible ways to interpret Theorems 9-4 and 9-5. Be careful, though. Write the converse of Theorem 9-4; if a line is a diameter, then it is also the perpendicular bisector of a chord. At first glance, students might think this is a true statement. Show students a counterexample or have a student come up and draw one (see Example 3). However, the converse of Theorem 9-5 is true; if the diameter bisects a chord and its corresponding arc, then the diameter is also perpendicular to the chord. When applying these theorems to a diagram, two of the following three things must be marked: chord is bisected, diameter passes through the center (to ensure that this chord is actually a diameter), or diameter is perpendicular to the chord. If two of these are marked, then it can be inferred that the third is also true.

Investigation 9-3 and Theorem 9-6 apply Theorems 9-3, 9-4, and 9-5 to two congruent chords in the same circle (or congruent circles). If you want, you can continue Investigation 9-3 on the same circle from Investigation 9-2. Draw a second chord that is the same length as \begin{align*}\overline{BC}\end{align*} (from Investigation 9-2) somewhere else in the circle. You can repeat step 2 from Investigation 9-2, rather than using step 2 from Investigation 9-3 as well. Both steps will produce the same result.

Review with students the definition of equidistant and why the shortest distance between a point and a line is the perpendicular line between them. In order for two chords to be congruent, using Theorem 9-6, the segments from the center to the chords must be marked congruent and perpendicular. Ask students if they notice any other properties of these segments. Students should notice that these segments are part of a diameter and that they also bisect each chord.

Finally, you can show students the following picture and see if they can find all the congruent chords, segments and arcs. Tell them that \begin{align*}\overline{BE} \cong \overline{IF}\end{align*}. This diagram applies all the theorems learned in this lesson.

\begin{align*}&\overline{BJ} \cong \overline{JE} \cong \overline{IM} \cong \overline{MF} && \widehat{BC} \cong \widehat{CE} \cong \widehat{HF} \cong \widehat{HI}\\ &\overline{CA} \cong \overline{AD} \cong \overline{AG} \cong \overline{AH} && \widehat{CD} \cong \widehat{HG}\\ &\overline{CJ} \cong \overline{MH} && \widehat{DE} \cong \widehat{GF}\\ &\overline{JK} \cong \overline{ML} && \widehat{CH} \cong \widehat{DG}\\ &\overline{KE} \cong \overline{LF}\\ &\overline{JA} \cong \overline{AM}\\ &\overline{AL} \cong \overline{AK}\\ &\overline{DK} \cong \overline{LG}\end{align*}

\begin{align*}*\end{align*}There are semicircles and major arcs that are also congruent.

And, \begin{align*}\Delta JKA \cong \Delta MLA\end{align*} by HL, SSS, or ASA

Inscribed Angles


This lesson will demonstrate how to find measures of inscribed angles and intercepted arcs.

Relevant Review

At this point, it would be very helpful to review all the theorems and vocabulary learned in this chapter. Make sure students have a firm grasp on the chapter up to this point. The chapter gets continually harder, so it is important that they have a strong foundation.

Double-check that every student has his/her flash cards of vocabulary and theorems and make sure they are using them regularly. One way to spot-test students on vocabulary is to have them lie out all their flash cards with the vocab or the name of the theorem side up. Then, you read the definition or theorem out loud. The first student to raise the correct card gets some sort of reward; either candy or an extra credit point. You can also tally the points and if any student reaches 5 (or any number of your choosing), then they will receive extra credit or a homework pass.

Teaching Strategies

It is very important that students understand how an inscribed angle and intercepted arc are defined and relate to each other. Any angle in a circle can (and usually does) have an intercepted arc, even though it is defined in terms of an inscribed angle. The important point to note is that the intercepted arc is the interior arc with the given endpoints on the circle. The blue arcs below are considered intercepted arcs for the inscribed angles below.

For Investigation 9-4, it might be helpful to already have three (or more) drawn inscribed angles on a handout for students. Pass these out and they still should draw in the corresponding central angle. Do not rush through this activity. Walk around to answer questions and let students arrive at the Inscribed Angle Theorem on their own. Then, proceed with lots of practice problems and examples to get students used to using this theorem.

Draw the pictures for Theorems 9-8 and 9-9 before telling the students what exactly the theorems are. For Theorem 9-8, ask students if they can conclude anything about \begin{align*}\angle ADB\end{align*} and \begin{align*}\angle ACB\end{align*}. For Theorem 9-9, ask students what the measure of the intercepted arc is, then they should be able to determine the measure of the inscribed angle. Here, the wording might be a little off from what is in the theorem. Make sure students can infer everything from these two theorems. First, in 9-8, the triangles created are only similar, not necessarily congruent. In 9-9, the endpoints of the inscribed angles are on a diameter and the diameter would be the hypotenuse of a right triangle.

Investigation 9-5 should be a teacher-led demonstration that can easily be done on an overhead projector (pre-cut a transparency into an inscribed quadrilateral and color the angles). Guide students towards the next theorem. The word “cyclic” is not used to describe these types of quadrilaterals. You can choose whether or not you would like to introduce this vocabulary or use what is in the text.

Angles of Chords, Secants, and Tangents


This lesson further explores angles in circles. Students will be able to find the measures of angles formed by chords, secants, and tangents.

Teaching Strategies

This lesson divides angles in circles into three different categories: angles with the vertex ON the circle, angles with the vertex IN the circle, and angles with the vertex OUTSIDE the circle.

central angle = intercepted arc

angle inside = half the sum of the intercepted arcs

angle on circle = half intercept arc

angle outside = half the difference of the intercepted arcs

Use the pictures above to help students generate formulas for each case. Then, draw the other options for angles on circles and angles outside circles.

Example 2 shows students that Theorem 9-8 (from the previous lesson) works for any angle where the vertex is on the circle.

Investigations 9-6, 9-7, and 9-8 can all be teacher-led investigations where students follow along and the class discovers the formulas together. Allow students to guess the possibilities for the formulas, even if they are wrong. Developing the correct formula will give students ownership over the material and it will help them retain the information.

Make sure you do plenty of practice problems in class to ensure that students are using the correct formula in the appropriate place. Give students a handout with several problems. You can choose to put the formulas on the board, let them use notes, or use nothing at all. Include problems from the previous lesson(s) as well.

Segments of Chords, Secants, and Tangents


This goal of this lesson is to explain the formulas for determining segment lengths formed by intersecting secants and tangents.

Relevant Review

Students might need a little algebraic review with solving quadratic equations. Problems involving tangents and secants can become a factoring problem or use the quadratic formula. Students might also need a review of square roots and simplifying square roots.

Teaching Strategies

Get students organized with all the information from this lesson and the previous two lessons. Have students draw and complete the following table.

Picture Angle Formula Segment Formula
\begin{align*}x^\circ = a^\circ\end{align*} Sides of angle are radii. No formula
\begin{align*}x^\circ = \frac{1}{2} (a^\circ + b^\circ)\end{align*} \begin{align*}pq = sr\end{align*}
\begin{align*}x^\circ = \frac{1}{2}a^\circ\end{align*} Sides are chords. No formula.
\begin{align*}x^\circ = \frac{1}{2}a^\circ\end{align*} One side is a chord, other is a ray. No formula.
\begin{align*}x^\circ = \frac{1}{2} (a^\circ - b^\circ)\end{align*} \begin{align*}s(s + p) = q(q + r)\end{align*}
\begin{align*}x^\circ = \frac{1}{2}(a^\circ - b^\circ)\end{align*} \begin{align*}s^2 = q(q + r)\end{align*}
\begin{align*}x^\circ = \frac{1}{2}(a^\circ - b^\circ)\end{align*} \begin{align*}s=q\end{align*}

Extension: Writing and Graphing the Equations of Circles

Show students how a point can be on a circle, using the Pythagorean Theorem. For example, if the equation of a circle is \begin{align*}x^2 + y^2 = 25\end{align*}, is (3, -4) on the circle? Yes. If students plug in 3 for \begin{align*}x\end{align*} and -4 for \begin{align*}y\end{align*}, they will see that the Pythagorean Theorem holds true. Conversely, if we tested (-6, 2), we would find that it is not on the circle because \begin{align*}36 + 4 \neq 25\end{align*}.

In Example 3, students might have problems finding the diameter. Remind them that the diameter is the longest chord in a circle. Students would need to count the squares vertically and horizontally to see where the longest segment would be (within the circle).

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