Medians in Triangles CK12 Editor Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required)
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www.ck12.org Concept 1. Medians in Triangles CONCEPT 1 Medians in Triangles Learning Objectives Construct the medians of a triangle. Apply the Concurrency of Medians Theorem to identify the point of concurrency of the medians of the triangle (the centroid). Use the Concurrency of Medians Theorem to solve problems involving the centroid of triangles. Introduction In our two last lessons we learned to circumscribe circles about triangles by finding the perpendicular bisectors of the sides and to inscribe circles within triangles by finding the triangle s angle bisectors. In this lesson we will learn how to find the location of a point within the triangle that involves the medians. Definition of Median of a Triangle A median of a triangle is the line segment that joins a vertex to the midpoint of the opposite side. Here is an example that shows the medians in an obtuse triangle. That the three medians appear to intersect in a point is no coincidence. As was true with perpendicular bisectors of the triangle sides and with angle bisectors, the three medians will be concurrent (intersect in a point). We call this point the centroid of the triangle. We can prove the following theorem about centroids. The Centroid of a Triangle Concurrency of Medians Theorem: The medians of a triangle will intersect in a point that is two-thirds of the distance from the vertices to the midpoint of the opposite side. Consider ABC with midpoints of the sides located at R, S, and T and the point of concurrency of the medians at the centroid, X. The theorem states that CX = 2 3 CS,AX = 2 3 AT, and BX = 2 3 BR. 1
www.ck12.org The theorem can be proved using a coordinate system and the midpoint and distance formulas for line segments. We will leave the proof to you (Homework Exercise #10), but will provide an outline and helpful hints for developing the proof. Example 1. Use The Concurrency of Medians Theorem to find the lengths of the indicated segments in the following triangle that has medians AT,CS, and BR as indicated. 1. If CS = 12, then CX = and XS =. CX = 2 3 12 = 8 XS = 1 3 12 = 4 2. If AX = 6. then XT = and AT =. We will start by finding AT. AX = 2 3 AT 6 = 2 3 AT 9 = AT Now for XT, XT = 1 3 AT = 1 3 9 = 3 2
www.ck12.org Concept 1. Medians in Triangles Napoleon s Theorem In the remainder of the lesson we will provide an interesting application of a theorem attributed to Napoleon Bonaparte, Emperor of France from 1804 to 1821, which makes use of equilateral triangles and centroids. We will explore Napoleon s theorem using The Geometer s Sketchpad. But first we need to review how to construct an equilateral triangle using circles. Consider XY and circles having equal radius and centered at X and Y as follows: Once you have hidden the circles, you will have an equilateral triangle. You can use the construction any time you need to construct an equilateral triangle by selecting the finished triangle and then making a Tool using the tool menu. Preliminary construction for Napoleon s Theorem: Construct any triangle ABC. triangle on each side. Construct an equilateral 3
www.ck12.org Find the centroid of each equilateral triangle and connect the centroids to get the Napoleon outer triangle. Measure the sides of the new triangle using Sketchpad. What can you conclude about the Napoleon outer triangle? (Answer: The triangle is equilateral.) 4
www.ck12.org Concept 1. Medians in Triangles This result is all the more remarkable since it applies to any triangle ABC. You can verify this fact in GSP by "dragging" a vertex of the original triangle ABC to form other triangles. The outer triangle will remain equilateral. Homework problem 9 will allow you to further explore this theorem. Example 2 Try this: a. Draw a triangle on a sheet of card stock paper (or thin cardboard) and locate the centroid. b. Carefully cut out the triangle. c. Hold your pencil point up and place the triangle on it so that the centroid rests on the pencil. d. What do you notice? The triangle balances on the pencil. Why does the triangle balance? Lesson Summary In this lesson we: Defined the centroid of a triangle. Stated and proved the Concurrency of Medians Theorem. Solved problems using the Concurrency of Medians Theorem. Demonstrated Napoleon s Theorem. Points to Consider So far we have been looking at relationships within triangles. In later chapters we will review the area of a triangle. When we draw the medians of the triangle, six smaller triangles are created. Think about the area of these triangles, and how that might relate to example 1 above. 5
www.ck12.org Review Questions 1. Find the centroid of ABC for each of the following triangles using Geometer s Sketchpad. For each triangle, measure the lengths of the medians and the distances from the centroid to each of the vertices. What can you conclude for each of the triangles? a. an equilateral triangle b. an isosceles triangle c. A scalene triangle 2. ABC has points R,S,T as midpoints of sides and the centroid located at point X as follows. Find the following lengths if XS = 10,XC =, CS =. 3. True or false: A median cannot be an angle bisector. Illustrate your reasoning with a drawing. 4. Find the coordinates of the centroid X of ABC with vertices A(2,3),B(4,1), and C(8,5). 5. Find the coordinates of the centroid X of ABC with vertices A(1,1),B(5,2), and C(6,6). Also, find XB. 6 6. Use the example sketch of Napoleon s Theorem to form the following triangle:
www.ck12.org Concept 1. Medians in Triangles a. Reflect each of the centroids in the line that is the closest side of the original triangle. b. Connect the points to form a new triangle that is called the inner Napoleon triangle. c. What can you conclude about the inner Napoleon triangle? 7. You have been asked to design a triangular metal logo for a club at school. Using the following rectangular coordinates, determine the logo s centroid. A(1, 0), B(1, 8),C(10, 4) 8. Prove Theorem 5-8. An outline of the proof together with some helpful hints is provided here. Proof. Consider ABC with A( p, 0), B(p, 0),C(x, y) as follows: Hints: Note that the midpoint of side AB is located at the origin. Construct the median from vertex C to the origin, and call it CO. The point of concurrency of the three medians will be located on CO at point P that is two-thirds of the way from C to the origin. 9. Prove: Each median of an equilateral triangle divides the triangle into two congruent triangles. Review Answers 1. a. Medians all have same length; distances from vertices to centroid all are same; they are two-thirds the lengths of the medians. b. Two of the medians have same length; distances from vertices to centroid are same for these two; all are two-thirds the lengths of the medians. c. Medians all have different lengths; distances from vertices to centroid; all are different; they are twothirds the lengths of the medians. 2. XC = 20,CS = 30. 3. False. The statement is true in the case of isosceles (vertex angle) and all angles in an equilateral triangle. 7
www.ck12.org 4. X( 14 3,3) a. X(4, 3) b. XB = 2. a. The triangle is equilateral. b. The difference in the areas of the inner and outer triangles is equal to the area of the original triangle. 5. The centroid will be located at X(4,4). The midpoint of the vertical side of the triangle is located at (1,4). Note that (1,4) is located 9 units from point C and that centroid will be one-third of the distance from (1,4) point to C. 8
www.ck12.org Concept 1. Medians in Triangles 6. a. Note that the midpoint of side AB is located at the origin. Construct the median from vertex C to the origin, and call it CO. The point of concurrency of the three medians will be located on CO at point P that is two-thirds of the way from C to the origin. b. Using slopes and properties of straight lines, the point can be determined to have coordinates P( 1 3 x, 1 3 y). c. Use the distance formula to show that point P is two-thirds of the way from each of the other two vertices to the midpoint of the opposite side. 7. Construct a median in an equilateral triangle. The triangles can be shown to be congruent by the SSS postulate. 9