<p>Students will work on problems involving segment lengths within circles. Students will:&nbsp;<br>- discover the segment product features of circles using technology.<br>- use the perpendicular property of tangents and the Pythagorean theorem to calculate missing segment lengths.<br>- use theorems and formulas involving secant, tangent, and chord segments to solve algebraic problems.<br>&nbsp;</p>

<p>- What are the different characteristics of circles, and how may they be utilized to solve problems?</p>

<p>- Chord: A line segment whose endpoints are on a circle.<br>- Circumscribed Polygon: A polygon such that every side of the polygon is tangent to the curve and that the curve is contained in the polygon.<br>- Converse of the Pythagorean Theorem: If in a triangle, <i>a</i>² + <i>b</i>² = <i>c</i>² and a, b, and c are the sides of the triangle, then the triangle is a right triangle; if <i>c</i>² &gt; <i>a</i>² + <i>b</i>², then the triangle is an obtuse triangle; if <i>c</i>² &lt; <i>a</i>² + <i>b</i>², then the triangle is an acute triangle.<br>- Inscribed Polygon: A polygon such that every vertex of the polygon is incident upon the curve and the polygon is contained inside the curve.<br>- Intersecting Chord Theorem: If two chords intersect in a circle, then the products of the lengths of the chords segments are equal.<br>- Perpendicular Lines: Two lines, segments, or rays that intersect to form right angles.<br>- Pythagorean Theorem: The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse; in any right triangle where the length of one leg is <i>a</i>, the length of the second leg is <i>b</i>, and the length of the hypotenuse is <i>c</i>, as in: <i>c</i>² = <i>a</i>² + <i>b</i>².<br>- Secant (of a circle): A line that intersects a circle in exactly two points.<br>- Secant Segment Theorem: If two secants intersect in the exterior of a circle, then the product of the measure of one secant segment and its external secant segment is equal to the product of the measures of the other secant and its external secant segment.<br>- Segment: A part of a line with two endpoints.<br>- Tangent (of a circle): A line that touches a circle in exactly one point. Tangent theorem 1 states that a line is tangent to a circle if and only if it is perpendicular to a radius drawn to the point of tangency. Tangent theorem 2 states that if two line segments from the same exterior point are tangent to the same circle, then they are congruent.<br>- Tangent Secant Segment Theorem: If a tangent and a secant intersect in the exterior of a circle, then the square of the measure of the tangent is equal to the product of the measures of the secant segment and its external secant segment.</p>

<p>- Lesson 3 PowerPoint presentation<br>- Printout of slides 3, 4, 5, 7, 8, 10, 11, 13, 15 for students from the Lesson 3 PowerPoint presentation<br>- Concept Builder worksheet</p>

<p>- Evaluate student performance in guided-practice examples by examining the correspondence between circle components and measurements, proper operations, and computational accuracy.</p>

<p>Scaffolding, Active Engagement, Modeling, Explicit Instruction<br>W: In this unit, students will learn about angle and arc relationships through various theorems. At this time, students may begin to wonder about segment lengths as well. This lesson investigates the secant, chord, and tangent segment relationships found in circles. Students may observe that certain diagrams look similar to problems encountered in Lessons 1 and 2, however the focus is on segments created rather than angles. This is an appropriate next step in the geometry learning process.&nbsp;<br>H: Lesson 3 engages students with technology-based activities. Introducing the lesson with a Technology Exploration will help to hook and sustain students' interest in the lesson themes. It also allows students to take personal responsibility for their learning.&nbsp;<br>E: This lesson aims to prepare students to complete similar ideas independently and use their knowledge in new situations. Using the PowerPoint presentation and handouts will help students stay organized and on track throughout the lesson. The use of extensions and critical-thinking tasks enables students to make sense of their learning and apply it in future learning situations.&nbsp;<br>R: Students modify their ideas depending on teacher feedback during instruction and technology activities. Feedback is essential for assisting students in determining where they need to improve their mental processes.&nbsp;<br>E: The Technology Exploration activities let students express their understanding and self-evaluate. As students advance through the activities, they learn why things aren't functioning and can make decisions based on their own progress. You should lead them down the right road in their learning process. Students will demonstrate their understanding in the Extension exercises near the end of the lesson.&nbsp;<br>T: This lesson offers two techniques for teaching the concepts. Choose the approach that best meets the demands of your classroom. Extension activities are designed to meet the demands of a wide range of school settings. The PowerPoint presentation and the Concept Builder worksheet allow students to gain experience with a range of learning techniques and can be useful when collaborating with different teachers/aides in the classroom.&nbsp;<br>O: The Technology Explorations option allows students to progress from investigative to independent learning, with minimum direct teaching. When using this strategy, make careful to help any students who miss out on some of the information. If proper checking for knowledge does not occur, students may have gaps in their understanding that will show up quickly when they attempt to do independent activities. Option 2 of the lesson uses a direct-instruction style to teaching, but allows students to practice and apply their knowledge through the same critical-thinking and extension activities.&nbsp;</p>

<p><strong>Option 1: Technology Approach</strong></p><p>Students can use technology to explain the important theorems of this lesson, including calculator explorations. Circle Product Theorems activities can be completed in groups, pairs, independently, or in class. Choose the method that best fits the needs of your students. The activities can be completed on either TI-84 or TI-Nspire™ calculators. Use the activity that matches the platform in your classroom. Use the links below to download all activity-related documents. When students are doing the exercise in class, be sure to answer questions and help them as needed.</p><p>TI 84 calculators: <a href="http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&amp;aId=12512.">http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&amp;aId=12512.</a></p><p>For TI-Nspire™ calculators, visit <a href="http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&amp;aId=12513.">http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&amp;aId=12513.</a></p><p>After completing the activity, students should be able to comprehend the theorems connected with segments formed by secants, chords, and tangents within a circle, as well as how they can be applied to solve problems. Use the PowerPoint presentation and example problems in Lesson 3 to fill in gaps in understanding and assess students' comprehension of the concepts. Ensure that students have the correct theorems and definitions in their Concept Builder worksheets, as well as the abilities required to apply the theorems to solve problems.</p><p><strong>Option 2: Direct Instruction</strong></p><p>Follow the Lesson 3 PowerPoint presentation to give students the information and examples they need to perform the concepts independently.</p><p>Lesson 3 starts with a discussion about tangent segments. Show students slides 2-5. Slide 2 explains the theorem about tangents. Allow students time to record this material on their Concept Builder worksheets. Before modeling the examples on slides 3-5, give students a printout of these slides. Printouts enable students to follow along with the modeling.</p><figure class="image"><img style="aspect-ratio:319/313;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_57.png" width="319" height="313"></figure><p>Example 1: Slide 3: <i>x</i> = \(\sqrt{93}\) or <i>x</i> ≈ 9.6</p><p><i>x</i>² + 14² = 17²<br><i>x</i>² + 196 = 289<br><i>x</i>² = 93<br><i>x</i> = \(\sqrt{93}\); or 9.6</p><p><strong>Note:</strong> This example needs students to memorize the Pythagorean theorem. Assist students in remembering this formula and explaining that because the tangent creates a perpendicular to the radius, a right triangle can be formed.</p><figure class="image"><img style="aspect-ratio:240/532;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_60.png" width="240" height="532"></figure><p>Example 2: Slide 4: <i>x</i> = 3</p><figure class="image"><img style="aspect-ratio:175/156;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_61.png" width="175" height="156"></figure><figure class="image"><img style="aspect-ratio:187/299;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_62.png" width="187" height="299"></figure><p>Example 3: Slide 5: x = \(\sqrt{89}\) or 9.4</p><p>5² + 8² = <i>x</i>²<br>25 + 64 = <i>x</i>²<br><i>x</i>² = 89<br><i>x</i> = \(\sqrt{89}\); or 9.4</p><p>Slides 6-8 continue the discussion of tangents. Display the theorem on slide 6, allowing students to capture and process the information offered. Provide students with printouts of slides 7 and 8, and then model these examples for them while they work on their worksheets.</p><figure class="image"><img style="aspect-ratio:308/280;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_64.png" width="308" height="280"></figure><p>Example 4: Slide 7: <i>x</i> = 4</p><figure class="image"><img style="aspect-ratio:307/262;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_65.png" width="307" height="262"></figure><p>Example 5: Slide 8: 40</p><figure class="image"><img style="aspect-ratio:298/246;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_66.png" width="298" height="246"></figure><p>Slides 9-11 describe the intersecting chord theorem. Display slide 9 for students to capture and process on the Concept Builder worksheet. Provide students with handouts of slides 10 and 11, and then demonstrate these instances while they work on their worksheets.</p><figure class="image"><img style="aspect-ratio:285/258;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_67.png" width="285" height="258"></figure><p>Example 6: Slide 10: \(20 \over 3\) or 6.67</p><p>9<i>x</i> = (12)5<br>9<i>x</i> = 60<br><i>x</i> = \(20 \over 3\); or 6.67</p><figure class="image"><img style="aspect-ratio:191/234;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_172.png" width="191" height="234"></figure><p>Example 7: Slide 11: <i>x</i> = \(1 \over 2\) or 0.5</p><p>3<i>x</i>(6) = 2(x + 4)<br>18<i>x</i> = 2x + 8<br>16<i>x</i> = 8<br><i>x</i> = \(1 \over 2\) or 0.5</p><figure class="image"><img style="aspect-ratio:196/228;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_173.png" width="196" height="228"></figure><p>Slides 12 and 13 deal with segments created by two secants intersecting a circle and then intersecting each other outside of the circle. Display the theorem on slide 12 so that students can copy it into their Concept Builder worksheets. Give students a printout of slide 13 to use while following the modeled example.</p><figure class="image"><img style="aspect-ratio:308/257;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_70.png" width="308" height="257"></figure><p>Example 8: Slide 13: <i>x</i> = 3</p><figure class="image"><img style="aspect-ratio:183/455;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_174.png" width="183" height="455"></figure><p>Example 8 requires students to factor a quadratic equation in order to solve the problem. Solving the quadratic equation reveals that two solutions are possible. Discuss with students which of the two solutions is the correct response to the provided problem. (In geometry, you cannot have negative lengths.)</p><p>Slides 14 and 15 are about segments created when a secant and a tangent intersect a circle and then intersect each other outside of the circle. Display the theorem on slide 14 so that students can copy it into their Concept Builder worksheets. Provide students with a printout of slide 15 to use as they follow along with the modeled example.</p><figure class="image"><img style="aspect-ratio:244/274;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_72.png" width="244" height="274"></figure><p>Example 9: Slide 15: <i>x</i> = 5.2</p><figure class="image"><img style="aspect-ratio:233/525;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_73.png" width="233" height="525"></figure><p><strong>Note: </strong>In Example 9, students must solve a quadratic equation using the quadratic formula to find the solution. Solving the quadratic equation reveals that two solutions are possible. Discuss with students which of the two solutions is the right one for the problem. (In geometry, you cannot have negative lengths.)</p><p><br><strong>Extension:</strong></p><p>Application problem: (See: <a href="https://whites-geometry-wiki.wikispaces.com/file/view/circle7.jpg/30631242/circle7.jpg">https://whites-geometry-wiki.wikispaces.com/file/view/circle7.jpg/30631242/circle7.jpg</a>)<br>The image below depicts an architectural structure that involves many circles and lines. From this perspective, two secant structures intersect a central circle. These have been traced with yellow and red. The chords within the circle measure 8 and 6 feet. If the longer pole measures 35 feet, calculate the length of the other pole. (Solution: <i>33.9 feet</i>).</p><p><strong>Note: </strong>The quadratic formula is required to solve this problem; one of the answers is negative and hence cannot be the solution.</p><figure class="image"><img style="aspect-ratio:288/219;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_74.png" width="288" height="219"></figure>