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6-1 Medians Objective: To identify and construct medians in triangles February 2011 Fellow: Brooke Odle Teacher: Ms. Sanchez Saint Vincent Academy Lesson Overview • Introduce topic – Vocabulary and Theorem 6-1 • Example Problems • Real World Application – Electrocardiograms • Class work – Clicker Questions What is a median? B median BD • In a triangle, a median is a segment that joins a vertex of the triangle and the midpoint of the side opposite that vertex. • A triangle has 3 medians. median TU C R median ZW Example 1 A • In ABC, CE and AD are medians. • Find BE if AB = 18. • ANSWER: 9 Example 2 A • In ABC, CE and AD are medians. • If CD = 2x+5, BD = 4x-1, and AE = 5x-2, find BE. • ANSWER: 13 Intersections of Medians • The medians of triangle K JKM, JR, KP, and MQ, Q intersect at a common point called the J centroid. R X • When 3 or more lines or P segments meet at the M same point, the lines X is the centroid of JKM are concurrent. and JR, KP, and MQ are all concurrent. Relationship between Lengths • There is a relationship between the length of the segment from the vertex to the centroid and the length of the segment from the centroid to the midpoint. C E A 13 26 D X F B AX = 26 XD = 13 What is the relationship between the length of AX and XD? Theorem 6-1 • The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint. 2x x Example 3 Y • Given XYZ, find YQ if QM = 4. • ANSWER = 8 N X Q R M Z Example 4 Y • Given XYZ and if QZ = 18, what is ZN? • ANSWER = 27 N X Q R M Z Real-World Application: Electrocardiogram Test • When a health professional wants to test the fitness of a person’s heart, they give an electrocardiogram (EKG or ECG) test. • An EKG test translates the heart’s electrical activity into line tracings on a paper called an electrocardiograph. – The spikes and dips on the line tracings are called waves Real-World Application: Electrocardiogram Test • Electrodes are placed on body (chest and both arms and legs)to record electrical activity of the heart. Real-World Application: Electrocardiogram Test You are a doctor. The cardiology technician gives you the results of your patient’s EKG test: – Equilateral triangle BCD is used to plot the EKG reading. – The patient has a left shoulder reading (S) of -1, a right shoulder reading (R) of 2, and a left leg reading (L) of 3. – You find the vertices of Triangle SRL (Einthoven’s triangle). Let’s construct the centroid P in Algodoo! Problem and graphic courtesy of McDougal Littell’s 2004 Geometry textbook Screenshot of Algodoo Scene: The purple bars represent the lines the students were to use to designate the medians for Triangle SRL Summary • Median: A segment in which one endpoint is the vertex of a triangle and the other is the midpoint of the side of the vertex. – Remember: A triangle has 3 medians. • Centroid: The point of intersection of the three medians of a triangle. • Concurrent: Three or more lines or segments that meet at a common point. • Theorem 6-1: The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint. Class Work CPS Clicker Questions Questions adapted from Kuta Software problems on Medians (http://www.kutasoftware.com/FreeWorksheets/Geo Worksheets/5-Medians.pdf) In Triangle GFE GT is a median. Find TE if FE = 8. G A. B. C. D. 16 10 4 8 [Default] [MC Any] [MC All] F T E In Triangle FGE, ET is a median. Find GF if TF = 6.3. A. B. C. D. 6.3 12.6 3.15 5 F T G [Default] [MC Any] [MC All] E In Triangle JIK, KL and IN are medians. Find LJ if IJ = 6. A. B. C. D. 12 4 3 6 [Default] [MC Any] [MC All] I L J N K P is the centroid of triangle JKL and PM = 8. Find JP. A. B. C. D. 16 4 12 8 [Default] [MC Any] [MC All] J N L P M K P is the centroid of triangle QRS and PT = 5. Find RT. A. B. C. D. R 5 10 8 15 [Default] [MC Any] [MC All] V U P S T Q