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1 Coach is the leader in standards-based, state-customized instruction for grades K 12 in English language arts, mathematics, science, and social studies. Our student texts deliver everything you need to meet your state standards and prepare your class for grade-level success! Coach lessons have just what you re looking for: Easy-to-follow, predictable lesson plans Florida Coach, Standards-Based Instruction, Geometry Standards-Based Curriculum Support! Focused instruction with modeled examples Guided practice with hints and support Higher-level thinking activities PLUS Chapter Reviews that target assessed skills Used by more students in the U.S. than any other state-customized series, Coach books are proven effective. Triumph Learning has been a trusted name in educational publishing for more than 40 years, and we continue to work with teachers and administrators to keep our books up to date improving test scores and maximizing student learning. Please visit our Web site for detailed product descriptions of all our instructional materials, including sample pages and more. Phone: (800) Fax: (866) customerservice@triumphlearning.com 188FL_Geo_HS_SE_Cvr.indd 1 188FL This book is printed on paper containing a minimum of 10% post-consumer waste. Developed in Consultation with Florida Educators 12/3/09 11:02:04 AM

2 Table of Contents Letter to the Student Test-Taking Checklist Florida Mathematics Standards Correlation Chart Benchmarks Chapter 1 Mathematical Reasoning and Logic Lesson 1 Introduction to Euclidean Geometry MA.912.G.8.1 Lesson 2 Converses, Inverses, and Contrapositives MA.912.D.6.2 Lesson 3 Geometric Proofs MA.912.D.6.4, MA.912.G.8.2, MA.912.G.8.4, MA.912.G.8.5 Lesson 4 Logical Equivalence and Validity MA.912.D.6.3, MA.912.D.6.4, MA.912.G.8.3 Chapter 1 Review Chapter 2 Points, Lines, Angles, and Planes Lesson 5 Distance and Midpoint MA.912.G.1.1 Lesson 6 Constructing Congruent Segments and Angles MA.912.G.1.2, MA.912.G.8.6 Lesson 7 Constructing Bisectors and Parallel and Perpendicular Lines MA.912.G.1.2, MA.912.G.4.2, MA.912.G.8.6 Lesson 8 Parallel Lines and Transversals MA.912.G.1.3 Chapter 2 Review Chapter 3 Polygons and Quadrilaterals Lesson 9 Polygons MA.912.G.2.1, MA.912.G.2.2 Lesson 10 Solving Problems with Congruent and Similar Polygons MA.912.G.2.3, MA.912.G.8.2 Lesson 11 Transformations, Congruence, Similarity, and Symmetry MA.912.G.2.4 Lesson 12 Tessellations MA.912.G.2.4 Lesson 13 Perimeter and Area of Polygons MA.912.G.2.5 Duplicating any part of this book is prohibited by law. 3

3 Lesson 14 Changes in the Dimensions of Polygons Lesson 15 Special Quadrilaterals Lesson 16 Proving Properties of Quadrilaterals Lesson 17 Proving Quadrilateral Theorems Chapter 3 Review MA.912.G.2.7 MA.912.G.3.1, MA.912.G.3.2 MA.912.G.3.3 MA.912.G.3.4 Chapter 4 Triangles Lesson 18 Classifying and Constructing Triangles Lesson 19 Points of Concurrency in Triangles Lesson 20 Constructing Congruent Triangles Lesson 21 Solving Problems with Congruent and Similar Triangles Lesson 22 Proving Congruence and Similarity Lesson 23 Triangle Inequality Theorems Lesson 24 Right Triangle Theorems Lesson 25 Pythagorean Theorem Lesson 26 Trigonometric Ratios and Special Right Triangles Chapter 4 Review MA.912.G.4.1, MA.912.G.8.6 MA.912.G.4.2, MA.912.G.8.6 MA.912.G.4.3, MA.912.G.8.6 MA.912.G.4.4, MA.912.G.4.5, MA.912.G.8.2 MA.912.G.4.5, MA.912.G.4.6 MA.912.G.4.7 MA.912.G.5.1, MA.912.G.5.2 MA.912.G.5.1, MA.912.G.5.4 MA.912.G.5.3, MA.912.G.5.4, MA.912.T.2.1 Chapter 5 Circles Lesson 27 Parts of a Circle Lesson 28 Measures of Arcs and Related Angles Lesson 29 Solving Real-World Problems with Circles Lesson 30 Equations of Circles Chapter 5 Review MA.912.G.6.2 MA.912.G.6.4 MA.912.G.6.5, MA.912.G.8.2 MA.912.G.6.6, MA.912.G Duplicating any part of this book is prohibited by law.

4 Table of Contents Chapter 6 Polyhedra and Other Solids Lesson 31 Polyhedra Lesson 32 Nets of Polyhedra Lesson 33 Spheres Lesson 34 Surface Area and Volume Lesson 35 Changes in the Dimensions of Solids Lesson 36 Congruent and Similar Solids Chapter 6 Review MA.912.G.7.1, MA.912.G.7.2 MA.912.G.7.1 MA.912.G.7.4 MA.912.G.7.5 MA.912.G.7.7 MA.912.G.7.6 Glossary Duplicating any part of this book is prohibited by law. 5

5 19 Points of Concurrency in Triangles MA.912.G.4.2, MA.912.G.8.6 If three or more lines are concurrent, then they intersect at the same point, which is called the point of concurrency. Knowing about four points of concurrency in a triangle can help you solve many different types of problems in mathematics and in the real world. The incenter is the point at which the angle bisectors of a triangle intersect. An angle bisector is a line or ray that divides an angle in half. The incenter, as its name suggests, is also the center of the circle that can be inscribed in the triangle. angle bisector incenter angle bisector angle bisector The orthocenter is the point at which the three altitudes of a triangle intersect. An altitude is the shortest distance between a vertex of a triangle and the opposite side (even if the opposite side needs to be extended to find that shortest distance). The shortest distance is always perpendicular to the opposite side. altitude altitude orthocenter altitude 132 Duplicating any part of this book is prohibited by law.

6 Lesson 19: Points of Concurrency in Triangles The circumcenter is the point at which the three perpendicular bisectors of the sides of a triangle intersect. A perpendicular bisector is a line that is perpendicular to a line segment and passes through the midpoint of the line segment. The circumcenter is also the center of the circle that could be circumscribed around the triangle such that all three vertices of the triangle are on the circle. Also, the circumcenter is equidistant (the same distance) from each of the triangle s vertices. perpendicular bisector perpendicular bisector circumcenter perpendicular bisector The centroid is the point at which the medians of a triangle intersect. The median of a triangle is a line segment drawn from a vertex of a triangle to the midpoint of the opposite side. The centroid divides each median into two segments, and the longer segment is twice as long as the shorter segment. The centroid is also the center of gravity of the triangle. midpoint midpoint centroid midpoint ou can construct each of these points of concurrency, using tools such as a compass. Duplicating any part of this book is prohibited by law. 133

7 EAMPLE 1 Construct the incenter of shown below. STRATEG Use a compass to draw an angle bisector for each vertex of the angle. Then find the point at which those angle bisectors intersect. STEP 1 Construct an angle bisector for vertex. Draw an arc from vertex so that the arc crosses both sides of the angle. Plot points where the arc crosses each side. Draw an arc from each of the points you plotted into the triangle s interior, using the same compass opening each time. Plot a point where the two arcs meet. Finally, draw a line segment from vertex through the intersection of the two arcs to the opposite side. This is an angle bisector. STEP 2 Using the steps described above, construct angle bisectors for vertices and. STEP 3 Label the point where the angle bisectors intersect as the incenter. Note: ou can inscribe a circle in the triangle, using the incenter as its center. incenter incenter SOLUTION The construction in Step 3 shows the incenter of. 134 Duplicating any part of this book is prohibited by law.

8 Lesson 19: Points of Concurrency in Triangles EAMPLE 2 A county wants to build a new public library. They want to build it at a location that is equidistant from the centers of three towns Derry, Evansville, and Franklinton shown on the map below. Which point of concurrency could you draw on this map to show the ideal location for the library? Derry Evansville Franklinton STRATEG STEP 1 STEP 2 Consider which point of concurrency is the same distance from all the vertices of a triangle. Construct this point. Determine which point of concurrency is equidistant from the vertices. The circumcenter of a triangle is the center of a circle which has all three vertices of the triangle on the circle. The distance from each vertex to the circumcenter is equal to a radius of the circle. Each radius is the same length, so each vertex will be equidistant from the circumcenter. Construct the circumcenter of the triangle. Construct the perpendicular bisector of the segment joining Derry and Evansville. Evansville Franklinton Derry Repeat this process for the other two sides. radii radius Evansville Franklinton Derry circumcenter SOLUTION The circumcenter of the triangle is the ideal location for the library. Duplicating any part of this book is prohibited by law. 135

9 EAMPLE 3 Point T is the centroid of MNP. If TR 6 cm, what is the length of MT? N Q T R M S P STRATEG STEP 1 STEP 2 SOLUTION Use what you know about medians and centroids. What do you know about medians and centroids? The centroid of a triangle divides each median into two segments. The longer segment is twice the length of the shorter segment. Find the length of MT. Centroid T divides the median MR into two segments. The shorter segment, TR, is 6 cm long, so the longer segment, MT, will be twice that length. MT cm The length of MT is 12 cm. COACHED EAMPLE In the diagram below, AH BC, BJ AC, and CK AB. What point of concurrency is point M? K M B H A J C THINKING IT THROUGH The line segment from the vertex of a triangle that is perpendicular to the opposite side is a(n) of the triangle. The three intersect at the point of concurrency called the. Since point M shows the intersection of AH, BJ, and CK, it is the of ABC. 136 Duplicating any part of this book is prohibited by law.

10 Lesson Practice Lesson 19: Points of Concurrency in Triangles Choose the correct answer. 1. In the diagram below, lines DE, FG, and HJ are perpendicular bisectors of the sides of ABC. A D B J K F H G Which describes point K? A. centroid B. circumcenter C. incenter D. orthocenter Use this diagram for questions 2 and 3. Point D is the centroid of KLM below. K E C 3. If MD 20 units, what is the length of DA? A. 10 units B. 20 units C. 30 units D. 40 units 4. Point is the orthocenter of TUV. U T W V Which term accurately describes T? A B C D altitude angle bisector median of triangle perpendicular bisector C D A M B L 2. If KC 22 units, what is the length of CM? A. 11 units B. 20 units C. 22 units D. 44 units Duplicating any part of this book is prohibited by law. 137

11 Use the map below for questions 5 and 6. Briarville Main Road Longview Road Avon Peach Road Claytown 6. A school is equidistant from each of the three towns shown on the map. Which could be used to find the school s location? A. centroid B. circumcenter C. incenter D. orthocenter 5. A museum is equidistant from each of the three roads shown on the map. Which could be used to find the museum s location? A. centroid B. circumcenter C. incenter D. orthocenter 138 Duplicating any part of this book is prohibited by law.