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Geometry A Unit 5: Constructions Lesson 4: Concurrency Properties of Triangles and It’s Applications Objective: Students will be able to investigate, justify, and apply theorems about the centroid, orthocenter, incenter, and circumcenter of a triangle. Vocabulary: Centroid Orthocenter Incenter Circumcenter Focus Questions: What are the points of concurrency of a triangle? How are these points used? Which type of triangle will have its incenter, orthocenter, circumcenter and centroid at the same point? (right, scalene, obtuse, equilateral) Game Plan: Do Now Quiz (5-1 & 5-2) Properties of Concurrency of Triangles Vocabulary Check Practice U-Try Focus Questions Homework: Finish U-Try Name: ______________________ Geometry A Ms. Torok Do Now: 1] Construct the following only using a straightedge and a compass. Perpendicular at P 2] Bisect A: Concurrency Properties of Triangles 1) Centroid 2) Incenter 3) Circumcenter 4) Orthocenter Lines that contain the same point are called concurrent. Concurrence is the concept of three or more lines intersecting in a single (common) point, having a single point of intersection. Every triangle has three perpendicular bisectors, three angle bisectors, three medians, and three altitudes. Etch- a- Sketch Sketch the 3 medians: Sketch the 3 angle bisectors: This point of intersection is called This point of intersection is called the __________ the __________ Concurrent Lines of a Triangle Centroid Circumcenter The medians of a triangle are Perpendicular bisectors of sides of a concurrent and intersect each other triangle are concurrent at a point in a ratio of 2:1. equidistant from the vertices. Incenter Orthocenter “Ortho” means “Right” The bisectors of the angles of a The point where the lines containing triangle meet at a point that is equally the altitudes are concurrent is called distant from the sides of the the orthocenter of a triangle. triangle. Vocabulary Check: Matching: A. Centroid ______ the point where all the altitudes meet B. Orthocenter ______the point where all the medians meet C. Circumcenter ______the point where all the angle bisectors meet D. Incenter ______the point where all the perpendicular bisectors of the sides meet Practice using the Concurrency Properties: I) Centroid: Theorem: The medians of a triangle are concurrent and intersect each other in a ratio of 2:1. Example #1: If point R is the centroid of triangle ABC, what is the perimeter of triangle ABC given that segments CF, DB, and AE are equal to 2, 3 and 4 respectively? Perimeter of triangle ABC = ________ Example #2: In the diagram below of ΔTEM, medians , , and intersect at D, and TB = 9. Find the length of . Answer: TD = ________ Example #3: Point Z is the centroid of triangle ABC and CZ = 18. What is the length of segment ? ZF = ________ Example #4: In the diagram of ΔABC below, Jose found centroid P by constructing the three medians. He measured and found it to be 6 inches. If PF = x, which equation can be used to find x? 1. x + x = 6 2. 2x + x = 6 3. 3x + 2x = 6 2 4. x + x = 6 3 Example #5: Point Z is the centroid of triangle ABC, ZF = 5, AD = 12 and BC = 18. What is the perimeter of triangle DCZ? Perimeter of triangle DCZ = ________ II) Incenter: Theorem: The bisectors of the angles of a triangle meet at a point that is equally distant from the sides of the triangle. Example #1: Which of the following points in the diagram below is the incenter of triangle ABC? Example #2: Given that point D is the incenter of isosceles triangle ABC, what is the measure of angle ADC? mADC = ________ Example #3: Given that point S is the incenter of right triangle PQR and angle RQS is 30°, what are the measures of angles RSQ and RPQ? mRSQ = ________ mRPQ = ________ III) Circumcenter: Theorem: Perpendicular bisectors of sides of a triangle are concurrent at a point equidistant from the vertices. Example #1: If circle O is circumscribed about triangle ABC, point O is what point of the triangle? 1. orthocenter 2. circumcenter 3. incenter 4. centroid Example #2: The diagram below shows the construction of the center of the circle circumscribed about ΔABC. This construction represents how to find the intersection of 1. the angle bisectors of ΔABC 2. the medians to the sides of ΔABC 3. the altitudes to the sides of ΔABC 4. the perpendicular bisectors of the sides of ΔABC IV) Orthocenter: Theorem: The point where the lines containing the altitudes are concurrent is called the orthocenter of a triangle. Example #1: Which point is the intersection of the altitudes of a triangle? 1] orthocenter 2] centroid 3] incenter 4] circumcenter Example #2: Which type of triangle would have its orthocenter on the triangle? 1] right 2] obtuse 3] scalene 4] equilateral U-Try: 1] Which is the point of intersection of the medians of a triangle? 1. orthocenter 2. centroid 3. incenter 4. circumcenter 2] Which of the four centers always remains on or inside a triangle? 1. incenter, only 2. incenter and centroid 3. orthocenter and incenter 4. circumcenter, only 3] Three or more lines that contain the same point are called: 1. parallel 2. perpendicular 3. current 4. concurrent 4] The incenter of a triangle can be located by finding the intersection of the: 1. altitudes 2. medians 3. perpendicular bisectors of the three sides 4. angle bisectors 5] The circumcenter of a triangle can be located by finding the intersection of the: 1. altitudes 2. medians 3. perpendicular bisectors of the three sides 4. angle bisectors