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Name:_________________________________________________ Date:________ Period:_______ Centroid Day 2: Notes Packet Do Now: Complete the chart Center of the Triangle Orthocenter At the concurrency of Centroid Located (inside, outside, on, etc…) Acute: Right: Obtuse: Always: Incenter Always: Circumcenter Acute: Right: Obtuse: All Of My Children Are Bringing In Create a circle? Peanut Butter Cookies Concurrency of the Medians: Recall: The ___________________ of a triangle is the line segment that joins the vertex to the ______________ of the opposite side of the triangle. The three medians of a triangle are concurrent in a point that is called the ___________________. There is a special relationship that involves the line segments when all of the three medians meet. The distance from each vertex to the centroid is ________________________ of the length of the entire median drawn from that vertex. Let’s Take a Look at the Diagram: AO = __________ CO = _________ BO = ________ In addition, the distance from each centroid to the opposite side (midpoint) is one-third of the distance of the entire median. OF= __________ OE = _________ OD = ________ The centroid also divides the median into two segments in the ratio 2:1, such that: = = = If you notice, the bigger part of the ratio is the segment that is drawn from the vertex to the centroid. The smaller part of the median is always the part that is drawn from the centroid to the midpoint of the opposite side. When working with these ratios, it is important to never mix the two up! It can sometimes be helpful to label the diagram with the ratios. Since the vertex to the centroid can be labeled as ______ and from the centroid to the midpoint of the opposite side can be labeled as ______, the entire median is known as _______. Therefore, whenever we do these centroid problems, we always set up a proportion. ̅̅̅̅ are concurrent at point Q. If TQ = 3x – 1 and TM = x + 1, what is the ̅̅̅̅̅ and 1.) In ∆, medians ̅̅̅̅̅? Justify your answer. length of median 2.) In ΔABC, points J, K, and L are midpoints of sides AB, BC, and AC, respectively. If the three medians of the triangle intersect at point P and the length of LP is 6, what is the length of BL? 3.) In triangle ABC, medians AD, BE, and CF are concurrent at point P. If AD = 24 inches, find the length of AP. 4.) In the diagram Jose found centroid P by constructing the three medians. He measured CF and found it to be 6 inches. If PF = x, then what equation can be used to find the value of x? (1) x + x = 6 (2) 2x + x = 6 (3) 3x + 2x = 6 2 (4) x + 3 = 6 5.) In ΔABC, medians CF and AD intersect at P. If CF = 6x and CP = 5x – 6, find x, CF, CP, and PF. Name:_________________________________________________ Date:________ Period:_______ Centroid: Day 2 Homework Complete the following examples. Show all work, including formulas. Make sure to justify all solutions. ̅̅̅̅̅ is a median of triangle PQR and point C is the centroid of triangle PQR. If 1.) In the diagram below, ̅̅̅̅̅. QC = 5x and CM = x + 12, determine and state the length of 2.) The three medians of a triangle intersect at a point. Which measurements could represent the segments of one of the medians? (1) 2 and 3 (2) 3 and 4.5 (3) 3 and 6 (4) 3 and 9 3.) In ΔRST, medians ̅̅̅̅̅ and ̅̅̅̅ are concurrent at point Q. If TQ = 9x–3, and QM = 3x+3, what is the ̅̅̅̅̅ length of median . 4.) In ∆ shown below, P is the centroid and BF = 18. What is the length of ̅̅̅̅ ? (1) 6 (2) 9 (3) 3 (4) 12 5.) In triangle ACE, ̅̅̅̅ is the median to ̅̅̅̅ , ̅̅̅̅ is the median to ̅̅̅̅ and ̅̅̅̅ is the median to ̅̅̅̅ . The ̅̅̅̅. medians meet at Q. If AQ = 48, find the measure of