CHAPTER 5 Geometry A/B Workbook NAME:___________________________________ TEACHER:_______________________________ 0 Table of Contents Date: Topic: Description: Page: 1 2 Geometry Section 5.1 Notes: Bisectors of Triangles Date: Question, Topics and Vocabulary Problems, Definitions and Work Segment Bisector: Perpendicular Bisector: Perpendicular Bisector Theorem: Converse of the Perpendicular Bisector Theorem: Example 1: a) Find the length of BC. What type of relationship should I use in order to find the length of BC? b) Find the length of XY. How is this diagram different to the previous diagram? 3 c) Find the length of PQ. Concurrent Lines: Point of Concurrency: Circumcenter Theorem: Example 2: A triangular-shaped garden is shown. Can a fountain be placed at the Where can the circumcenter and still be inside the garden? circumcenter be located at? Angle Bisector: 4 Angle Bisector Theorem: Converse of the Angle Bisector Theorem: Example 3: a) Find the length of DB. What conclusion can be made if given a segment is an angle bisector? b) Find mWYZ. What conclusions can be made if given the two segments from the angle bisector are equal and intersect at a right angle? c) Find the length of QS. Given the set up to the right, what can I do to my two expressions? And why? 5 Incenter: Incenter Theorem: Example 4: Use the diagram to the right. a) Find ST if S is the incenter of ΔMNP. If S is the incenter, what do I know about my triangle? b) Find mSPU if S is the incenter of ΔMNP. Summary 1) Find the value of x. 2) Find the length of KL. 3) Find 4) Point A is the incenter of . Find the measure of . 6 Geometry Name: _____________________________________ Section 5.1 Worksheet For numbers 1 – 6, find each measure. 1. TP 2. VU 3. KN 4. NJZ 5. QA 6. MFZ 7 For numbers 7 & 8, point L is the circumcenter of ABC. List any segment(s) congruent to each segment. 7. BN 8. BL For numbers 9 & 10, point A is the incenter of PQR. Find each measure. 9. YLA 10. YGA 8 Geometry Section 5.2 Notes: Medians and Altitudes of Triangles Date: Question, Topics and Problems, Definitions and Work Vocabulary Median: Centroid: Centroid Theorem: Example 1: In ΔXYZ, P is the centroid and YV = 12. Find YP and PV. Given the information about triangle XYZ, how can I mark up the triangle if knowing P is the centroid? Example 2: In ΔABC, CG = 4. Find GE. Given the picture of the triangle, what is G in my picture? 9 Altitude: Orthocenter: Point of Concurrency : Special Segments and Points in Triangles Special Property: Perpendicular Bisector Is there an acronym I could create to help me memorize these special segments/points in Point of Concurrency : triangles? Special Property: Angle Bisector 10 Median Is there an acronym I could Special Property: create to help me memorize these Point of Concurrency : special segments/points in triangles? Point of Concurrency : Special Property: Altitude Summary 5.2 Exit Slip In , NQ = 6, RK = 3, and PK = 4. Find each measure using the figure below. 1) KM 2) KQ 3) LK 4) LR 5) NK 6) PM 11 7. Draw an altitude from point B to ̅̅̅ ̅ in triangle ABC. Make all necessary markings. For numbers 8 - 11, give the name of the point of concurrency for each of the following. 8. Angle Bisectors of a Triangle 9. Medians of a Triangle 10. Altitudes of a Triangle 11. Perpendicular Bisectors of a Triangle For numbers 12 – 13, complete each of the following statements. 12. The incenter of a triangle is equidistant from the _______________ of the triangle. 13. The circumcenter of a triangle is equidistant from the _________________ of the triangle. 12 Geometry Name: _____________________________________ Section 5.2 Worksheet For numbers 1 – 6, in ABC, CP = 30, EP = 18, and BF = 39. Find each measure. 1. PD 2. FP 3. BP 4. CD 5. PA 6. EA For numbers 7 – 12, in MIV, Z is the centroid, MZ = 6, YI = 18, and NZ = 12. Find each measure. 7. ZR 8. YZ 9. MR 10. ZV 11. NV 12. IZ 13. DISTANCES For what kind of triangle is there a point where the distance to each side is half the distance to each vertex? Explain. 14. MEDIANS Look at the right triangle below. What do you notice about the orthocenter and the vertices of the triangle? 13 14 Geometry Section 5.3 Notes: Inequalities in One Triangle Date: Question, Topics and Vocabulary Problems, Definitions and Work Definition of Inequality Exterior Angle Inequality Example 1: Use the diagram to the right. a) Use the Exterior Angle Inequality Theorem to list all angles whose measures are less than m14. b) Use the Exterior Angle Inequality Theorem to list all angles whose measures are greater than m5. Longest Side Vs. Longest Angle Smallest Side Vs. Shortest Side Angle-Side Relationships in Triangles (5.9) 15 Angle- Side Relationships in Triangles (5.10) Example 2: List the angles of ΔABC in order from smallest to largest. How do I know which side corresponds to which angle? Example 3: List the sides of ΔABC in order from shortest to longest. Example 4: HAIR ACCESSORIES Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the dimensions shown, which two ends should she tie? How do I know which angle corresponds to which side? Example 5: List the angles of each triangle in order from smallest to largest. Example 6: List the angles of each triangle in order from smallest to largest. 16 Example 7: List the angles of each triangle in order from smallest to largest. ∆XYZ, where XY = 25, YZ = 11, and XZ = 15 Example 8: List the sides of each triangle in order from shortest to longest. Example 9: List the sides of each triangle in order from shortest to longest. 5.1-5.3 Vocab Skills Match each word to the sentence in which it best fills in the blank. Write your answer on the line to the left. Check _____1) The point of intersection of the angle bisectors in a triangle is called the_?_. Answer Choices 12 _____2) The _?_ divides the medians in a triangle into the and parts. A. circumcenter 33 B. incenter C. sides _____3) The centroid is the intersection of the_?_in a triangle. D. vertices E. medians F. orthocenter _____4) The circumcenter is equidistant from the _?_in a triangle. G.centroid _____5) The incenter is equidistant from the_?_in a triangle _____6) The point of intersection of the perpendicular bisectors in a triangle is called the_?_. _____7) The altitudes in a triangle intersect at a point called the _?_. 17 Summary 1. a) List the sides from biggest to smallest. Write your answer as an inequality. B 20 E 110 M b) List the angles from largest to smallest. Write your answer as an inequality. A 25 12 W 15 P 18 Geometry Name: _____________________________________ Section 5.3 Worksheet For numbers 1 – 4, use the figure at the right to determine which angle has the greatest measure. 1. 1, 3, 4 2. 4, 8, 9 3. 2, 3, 7 4. 7, 8, 10 For numbers 5 – 8, use the Exterior Angle Inequality Theorem to list all angles that satisfy the stated condition. 5. measures are less than m1. 6. measures are less than m3. 7. measures are greater than m7. 8. measures are greater than m∠2. For numbers 9 – 12, use the figure at the right to determine the relationship between the measures of the given angles. 9. mQRW, mRWQ 10. mRTW, mTWR 11. mRST, mTRS 12. mWQR, mQRW 19 Use the figure at the right to determine the relationship between the lengths of the given sides. 13. DH, GH 14. DE, DG 15. EG, FG 16. DE, EG 17. SPORTS The figure shows the position of three trees on one part of a disc golf course. At which tree position is the angle between the trees the greatest 20 Geometry Section 5.5 Notes: The Triangle Inequality Date: Question, Topics and Vocabulary Problems, Definitions and Work Triangle Inequality Theorem Example 1: a) Is it possible to form a triangle with side lengths of 6.5, 6.5, and 14.5? If not, explain why Is this triangle not. possible? b) Is it possible to form a triangle with side lengths of 6.8, 7.2, 5.1? If not, explain why not. Example 2: In ΔPQR, PQ = 7.2 and QR = 5.2. Which measure cannot be PR? a) 7 b) 9 c) 11 d) 13 Example 3: TRAVEL The towns of Jefferson, Kingston, and Newbury are shown in the map below. Prove that driving first from Jefferson to Kingston and then Kingston to Newbury is a greater distance than driving from Jefferson to Newbury. 21 Example 4:The lengths of two sides of a triangle are given. Find the Range of the lengths: range of possible lengths for the third side. a. 4, 8 b. 13, 8 c. 10, 15 Example 5: Error Analysis A student draws a triangle with a perimeter of 12 in. The student says that the longest side measures 7 in. How do you know that the student is incorrect? Explain. Summary 1. Can you make a triangle out of the following lengths: 1 cm, 3 cm, and 8 cm? Why or why not? 2.