6 segment from vertex A to BC. . Label the endpoint D. is an altitude of ABC. 4 b. Construct the altitudes to the other two sides of ABC.
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1 6. Medians and ltitudes of Triangles ssential uestion What conjectures can you make about the medians and altitudes of a triangle? inding roperties of the Medians of a Triangle Work with a partner. Use dynamic geometry software. raw any. a. lot the midpoint of and label it. raw, which is a median of. onstruct the medians to the other two sides of G medians Sample oints (1, 4) (6, 5) (8, 0) (7,.5) (4.5, ) G(5, ) b. What do you notice about the medians? rag the vertices to change. Use your observations to write a conjecture about the medians of a triangle. c. In the figure above, point G divides each median into a shorter segment and a longer segment. ind the ratio of the length of each longer segment to the length of the whole median. Is this ratio always the same? Justify your answer. LOOKING OR STRUTUR To be proficient in math, you need to look closely to discern a pattern or structure. inding roperties of the ltitudes of a Triangle Work with a partner. Use dynamic geometry software. raw any. a. onstruct the perpendicular 6 segment from vertex to. Label the endpoint. 5 is an altitude of. altitude 4 b. onstruct the altitudes to the other two sides of. What do you notice? c. Write a conjecture about the altitudes of a triangle. Test your 1 conjecture by dragging the 0 vertices to change. ommunicate Your nswer 0 1. What conjectures can you make about the medians and altitudes of a triangle? 4. The length of median RU in RST is inches. The point of concurrency of the three medians of RST divides RU into two segments. What are the lengths of these two segments? Section 6. Medians and ltitudes of Triangles 19
2 6. Lesson What You Will Learn ore Vocabulary median of a triangle, p. 0 centroid, p. 0 altitude of a triangle, p. 1 orthocenter, p. 1 revious midpoint concurrent point of concurrency Use medians and find the centroids of triangles. Use altitudes and find the orthocenters of triangles. Using the Median of a Triangle median of a triangle is a segment from a vertex to the midpoint of the opposite side. The three medians of a triangle are concurrent. The point of concurrency, called the centroid, is inside the triangle. Theorem Theorem 6.7 entroid Theorem The centroid of a triangle is two-thirds of the distance from each vertex to the midpoint of the opposite side. The medians of meet at point, and =, =, and =. roof igideasmath.com inding the entroid of a Triangle Use a compass and straightedge to construct the medians of. Step 1 Step Step ind midpoints raw. ind the midpoints of,, and. Label the midpoints of the sides,, and, respectively. raw medians raw,, and. These are the three medians of. Label a point Label the point where,, and intersect as. This is the centroid. Using the entroid of a Triangle S U 8 V R W T In RST, point is the centroid, and S = 8. ind W and SW. S = SW entroid Theorem 8 = SW Substitute 8 for S. 1 = SW Multiply each side by the reciprocal,. Then W = SW S = 1 8 = 4. So, W = 4 and SW = 1. 0 hapter 6 Relationships Within Triangles
3 INING N NTRY OINT The median SV is chosen in xample because it is easier to find a distance on a vertical segment. JUSTIYING ONLUSIONS You can check your result by using a different median to find the centroid. RING In the area formula for a triangle, = 1 bh, you can use the length of any side for the base b. The height h is the length of the altitude to that side from the opposite vertex. inding the entroid of a Triangle ind the coordinates of the centroid of RST with vertices R(, 1), S(5, 8), and T(8, ). Step 1 Graph RST. Step Use the Midpoint ormula to find the midpoint V of RT and sketch median SV. V ( + 8, 1 + ) = (5, ) Step ind the centroid. It is two-thirds of the distance from each vertex to the midpoint R(, 1) of the opposite side. 4 The distance from vertex S(5, 8) to V(5, ) is 8 = 6 units. So, the centroid is (6) = 4 units down from vertex S on SV. So, the coordinates of the centroid are (5, 8 4), or (5, 4). Monitoring rogress There are three paths through a triangular park. ach path goes from the midpoint of one edge to the opposite corner. The paths meet at point. 1. ind S and when S = 100 feet.. ind T and when T = 1000 feet.. ind and T when T = 800 feet. Help in nglish and Spanish at igideasmath.com ind the coordinates of the centroid of the triangle with the given vertices. 4. (, 5), G(4, 9), H(6, 1) 5. X(, ), Y(1, 5), Z( 1, ) Using the ltitude of a Triangle n altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side y altitude from to R R S(5, 8) (5, 4) T(8, ) V(5, ) x S R T R ore oncept Orthocenter The lines containing the altitudes of a triangle are concurrent. This point of concurrency is the orthocenter of the triangle. The lines containing,, and meet at the orthocenter G of. G Section 6. Medians and ltitudes of Triangles 1
4 s shown below, the location of the orthocenter of a triangle depends on the type of triangle. RING The altitudes are shown in red. Notice that in the right triangle, the legs are also altitudes. The altitudes of the obtuse triangle are extended to find the orthocenter. cute triangle is inside triangle. Right triangle is on triangle. inding the Orthocenter of a Triangle Obtuse triangle is outside triangle. ind the coordinates of the orthocenter of XYZ with vertices X( 5, 1), Y(, 4), and Z(, 1). Step 1 Graph XYZ. y x = Step ind an equation of the line that contains the altitude from Y to XZ. ecause XZ 5 is Y (, ) horizontal, the altitude is vertical. The y = x + 4 line that contains the altitude passes through Y(, 4). So, the equation of 1 the line is x =. 1 1 x Step ind an equation of the line that contains the altitude from X to YZ. slope of YZ = 1 4 ( ) = 1 X Z ecause the product of the slopes of two perpendicular lines is 1, the slope of a line perpendicular to YZ is 1. The line passes through X( 5, 1). y = mx + b Use slope-intercept form. 1 = 1( 5) + b Substitute 1 for y, 1 for m, and 5 for x. 4 = b Solve for b. So, the equation of the line is y = x + 4. Step 4 ind the point of intersection of the graphs of the equations x = and y = x + 4. Substitute for x in the equation y = x + 4. Then solve for y. y = x + 4 Write equation. y = + 4 Substitute for x. y = Solve for y. So, the coordinates of the orthocenter are (, ). Monitoring rogress Help in nglish and Spanish at igideasmath.com Tell whether the orthocenter of the triangle with the given vertices is inside, on, or outside the triangle. Then find the coordinates of the orthocenter. 6. (0, ), (0, ), (6, ) 7. J(, 4), K(, 4), L(5, 4) hapter 6 Relationships Within Triangles
5 oncept Summary In an isosceles triangle, the perpendicular bisector, angle bisector, median, and altitude from the vertex angle to the base are all the same segment. In an equilateral triangle, this is true for any vertex. roving a roperty of Isosceles Triangles rove that the median from the vertex angle to the base of an isosceles triangle is an altitude. Given is isosceles, with base. is the median to base. rove is an altitude of. aragraph roof Legs and of isosceles are congruent. because is the median to. lso, by the Reflexive roperty of ongruence (Thm..1). So, by the SSS ongruence Theorem (Thm. 5.8). because corresponding parts of congruent triangles are congruent. lso, and are a linear pair. and intersect to form a linear pair of congruent angles, so and is an altitude of. Monitoring rogress Segments, Lines, Rays, and oints in Triangles perpendicular bisector Help in nglish and Spanish at igideasmath.com 8. WHT I? In xample 4, you want to show that median is also an angle bisector. How would your proof be different? xample oint of oncurrency roperty xample circumcenter The circumcenter of a triangle is equidistant from the vertices of the triangle. angle bisector incenter The incenter I of a triangle is equidistant from the sides of the triangle. I median centroid The centroid R of a triangle is two thirds of the distance from each vertex to the midpoint of the opposite side. R altitude orthocenter The lines containing the altitudes of a triangle are concurrent at the orthocenter O. O Section 6. Medians and ltitudes of Triangles
6 6. xercises ynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck 1. VOULRY Name the four types of points of concurrency. Which lines intersect to form each of the points?. OMLT TH SNTN The length of a segment from a vertex to the centroid is the length of the median from that vertex. Monitoring rogress and Modeling with Mathematics In xercises 6, point is the centroid of LMN. ind N and. (See xample 1.). N = 9 4. N = 1 M L 5. N = 0 6. N = 4 L N M N In xercises 7 10, point is the centroid of. ind and. N M M 7. = 5 8. = 11 N L L In xercises 11 14, point G is the centroid of. G = 6, = 1, and = 15. ind the length of the segment G 14. G 6 G In xercises 15 18, find the coordinates of the centroid of the triangle with the given vertices. (See xample.) 15. (, ), (8, 1), (5, 7) 16. (1, 5), G(, 7), H( 6, ) 17. S(5, 5), T(11, ), U( 1, 1) 18. X(1, 4), Y(7, ), Z(, ) In xercises 19, tell whether the orthocenter is inside, on, or outside the triangle. Then find the coordinates of the orthocenter. (See xample.) 9. = = L(0, 5), M(, 1), N(8, 1) 0. X(, ), Y(5, ), Z(, 6) 1. ( 4, 0), (1, 0), ( 1, ). T(, 1), U(, 1), V(0, 4) ONSTRUTION In xercises 6, draw the indicated triangle and find its centroid and orthocenter.. isosceles right triangle 4. obtuse scalene triangle 5. right scalene triangle 6. acute isosceles triangle 4 hapter 6 Relationships Within Triangles
7 RROR NLYSIS In xercises 7 and 8, describe and correct the error in finding. oint is the centroid of = = (18) = 1 = = (4) = 16 ROO In xercises 9 and 0, write a proof of the statement. (See xample 4.) 9. The angle bisector from the vertex angle to the base of an isosceles triangle is also a median. 0. The altitude from the vertex angle to the base of an isosceles triangle is also a perpendicular bisector. RITIL THINKING In xercises 1 6, complete the statement with always, sometimes, or never. xplain your reasoning. 1. The centroid is on the triangle.. The orthocenter is outside the triangle.. median is the same line segment as a perpendicular bisector. 4. n altitude is the same line segment as an angle bisector. 5. The centroid and orthocenter are the same point. 6. The centroid is formed by the intersection of the three medians. 7. WRITING ompare an altitude of a triangle with a perpendicular bisector of a triangle. 8. WRITING ompare a median, an altitude, and an angle bisector of a triangle. = 4 = MOLING WITH MTHMTIS ind the area of the triangular part of the paper airplane wing that is outlined in red. Which special segment of the triangle did you use? in. 9 in. 40. NLYZING RLTIONSHIS opy and complete the statement for with centroid K and medians H, J, and G. a. J = KJ b. K = KH c. G = K d. KG = G MTHMTIL ONNTIONS In xercises 41 44, point is the centroid of. Use the given information to find the value of x. G 41. = 4x + 5 and = 9x 4. G = x 8 and G = x + 4. = 5x and = x 44. = 4x 1 and = 6x MTHMTIL ONNTIONS Graph the lines on the same coordinate plane. ind the centroid of the triangle formed by their intersections. y 1 = x 4 y = 4 x + 5 y = x RITIL THINKING In what type(s) of triangles can a vertex be one of the points of concurrency of the triangle? xplain your reasoning. Section 6. Medians and ltitudes of Triangles 5
8 47. WRITING UTIONS Use the numbers and symbols to write three different equations for = HOW O YOU S IT? Use the figure. K h J N M 9 L 9 a. What type of segment is KM? Which point of concurrency lies on KM? b. What type of segment is KN? Which point of concurrency lies on KN? c. ompare the areas of JKM and KLM. o you think the areas of the triangles formed by the median of any triangle will always compare this way? xplain your reasoning. 49. MKING N RGUMNT Your friend claims that it is possible for the circumcenter, incenter, centroid, and orthocenter to all be the same point. o you agree? xplain your reasoning. 50. RWING ONLUSIONS The center of gravity of a triangle, the point where a triangle can balance on the tip of a pencil, is one of the four points of concurrency. raw and cut out a large scalene triangle on a piece of cardboard. Which of the four points of concurrency is the center of gravity? xplain. 51. ROO rove that a median of an equilateral triangle is also an angle bisector, perpendicular bisector, and altitude. 5. THOUGHT ROVOKING onstruct an acute scalene triangle. ind the orthocenter, centroid, and circumcenter. What can you conclude about the three points of concurrency? 5. ONSTRUTION ollow the steps to construct a nine-point circle. Why is it called a nine-point circle? Step 1 onstruct a large acute scalene triangle. Step ind the orthocenter and circumcenter of the triangle. Step ind the midpoint between the orthocenter and circumcenter. Step 4 ind the midpoint between each vertex and the orthocenter. Step 5 onstruct a circle. Use the midpoint in Step as the center of the circle, and the distance from the center to the midpoint of a side of the triangle as the radius. 54. ROO rove the statements in parts (a) (c). Given L and M are medians of scalene LMN. oint R is on L such that L R. oint S is on M such that M S. rove a. NS NR b. NS and NR are both parallel to LM. c. R, N, and S are collinear. Maintaining Mathematical roficiency etermine whether is parallel to. (Section.5) 55. (5, 6), ( 1, ), ( 4, 9), ( 16, ) 56. (, 6), (5, 4), ( 14, 10), (, 7) 57. (6, ), (5, ), ( 4, 4), ( 5, ) 58. ( 5, 6), ( 7, ), (7, 1), (4, 5) Reviewing what you learned in previous grades and lessons 6 hapter 6 Relationships Within Triangles
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