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5-2 Medians and Altitudes of Triangles COORDINATE GEOMETRY Find the coordinates of the centroid of each triangle with the given vertices. 11. A(–1, 11), B(3, 1), C(7, 6) 12. X(5, 7), Y(9, –3), Z(13, 2) SOLUTION: The midpoint D of SOLUTION: The midpoint D of is Note that is a line that is Note that is a line that connects the vertex C and D, the midpoint of . The distance from D(1, 6) to C(7, 6) is 7 – 1 or 6 units. If P is the centroid of the triangle ABC, then So, the centroid is connects the vertex Z and D, the midpoint of . The distance from D(7, 2) to Z(13, 2) is 13 – 7 or 6 units. If P is the centroid of the triangle XYZ, then So, the centroid is or 4 units to the left of Z. The coordinates of the centroid(P) are (13– 4, 2) or (9, 2). or 4 units to the left of C. The coordinates of the centroid (P) are (7 – 4, 6) or (3, 6). COORDINATE GEOMETRY Find the coordinates of the orthocenter of each triangle with the given vertices. 14. J(3, –2), K(5, 6), L(9, –2) SOLUTION: 12. X(5, 7), Y(9, –3), Z(13, 2) The slope of is or So, the slope of SOLUTION: The midpoint D of the altitude, which is perpendicular to is Note that is a line that connects the vertex Z and D, the midpoint of . The distance from D(7, 2) to Z(13, 2) is 13 – 7 or 6 units. If P is the centroid of the triangle XYZ, then So, the centroid is or 4 units to the left of Z. The coordinates of the centroid(P) are (13– 4, 2) or (9, 2). is Now, the equation of the altitude from L to is: Use the same method to find the equation of the altitude from J to .That is, Solve the equations to find the intersection point of the altitudes. eSolutions Manual - Powered by Cognero Page 1 5-2 Medians and Altitudes of Triangles COORDINATE GEOMETRY Find the coordinates of the orthocenter of each triangle with the given vertices. 14. J(3, –2), K(5, 6), L(9, –2) SOLUTION: The slope of is or So, the slope of the altitude, which is perpendicular to 15. R(–4, 8), S(–1, 5), T(5, 5) SOLUTION: The slope of is or –1. So, the slope of the altitude, which is perpendicular to the equation of the altitude from T to is 1. Now, is: is Now, the equation of the altitude from L to Use the same way to find the equation of the altitude is: from R to .That is, Solve the equations to find the intersection point of the altitudes. Use the same method to find the equation of the altitude from J to .That is, So, the coordinates of the orthocenter of 4, –4). is (– Solve the equations to find the intersection point of the altitudes. ALGEBRA Use the figure. So, the coordinates of the orthocenter of –1). is (5, 24. If m is an altitude of 2 = 3x + 13, find m , m 1 = 2x + 7,and 1 and m 2. SOLUTION: By the definition of altitude, Substitute 14 for x in 15. R(–4, 8), S(–1, 5), T(5, 5) eSolutions Manual - Powered by Cognero SOLUTION: Page 2 is not an altitude of 5-2 Medians and Altitudes of Triangles because m ECA = 92. ALGEBRA Use the figure. 24. If m is an altitude of 2 = 3x + 13, find m CCSS ARGUMENTS Use the given information to determine whether is a perpendicular bisector, median, and/or an altitude of . , m 1 = 2x + 7,and 1 and m 2. SOLUTION: By the definition of altitude, 27. SOLUTION: is an altitude by the definition Since , of altitude.We don't know if it is a perpendicular bisector because it is not evident that M is the midpoint of Substitute 14 for x in 28. SOLUTION: Since , we know that by CPCTC. Since and they are a linear pair, then we know they are right angles and . Therefore, is the perpendicular bisector, median, and altitude of . 25. Find the value of x if AC = 4x – 3, DC = 2x + 9, m ECA = 15x + 2, and . is a median of Is ? Explain. also an altitude of 29. SOLUTION: Since we know that midpoint of . Therefore, . SOLUTION: Given: AC = DC. 30. then M is the is the median of and SOLUTION: Since Substitute 6 for x in m is not an altitude of and , we can prove by HL. Therefore, we know that by CPCTC, making M the midpoint of . Therefore, is the perpendicular bisector, median, and altitude of . ECA. because m ECA = 92. CCSS ARGUMENTS Use the given information to determine eSolutions Manual - Powered by Cognerowhether is a perpendicular bisector, median, and/or an altitude of . Page 3