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NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 5-2 Practice Medians and Altitudes of Triangles Medians A median is a line segment that connects a vertex of a triangle to the midpoint of the opposite side. The three medians of a triangle intersect at the centroid of the triangle. The centroid is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. Example: In △ABC, U is the centroid and BU = 16. Find UK and BK. 2 3 BU = BK 2 3 16 = BK 24 = BK BU + UK = BK 16 + UK = 24 UK = 8 Exercises: In △ABC, AU = 16, BU = 12, and CF = 18. Find each measure. 1. UD 2. EU 3. CU 4. AD 5. UF 6. BE In △CDE, U is the centroid, UK = 12, EM = 21, and UD = 9. Find each measure. 7. CU 8. MU 9. CK 10. JU 11. EU 12. JD Chapter 5 11 Glencoe Geometry NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 5-2 Practice (cont.) Medians and Altitudes of Triangles Altitudes An altitude of a triangle is a segment from a vertex to the line containing the opposite side meeting at a right angle. Every triangle has three altitudes which meet at a point called the orthocenter. Example: The vertices of △ABC are A(1, 3), B(7, 7) and C(9, 3). Find the coordinates of the orthocenter of △ABC. Find the point where two of the three altitudes intersect. ���� . Find the equation of the altitude from A to 𝐵𝐵 Find the equation of the altitude from C to ����� 𝐴𝐴 . ���� has a slope of −2, then the altitude If 𝐵𝐵 1 has a slope of . 2 y – 𝑦1 = m(x – 𝑥1 ) 3 2 of – . y – 𝑦1 = m(x – 𝑥1 ) Point-slope form 1 y – 3 = (x – 1) 2 1 1 y–3= x– 2 2 1 5 y= x+ 2 2 1 m = , (𝑥1 , 𝑦1 ) = A(1, 3) y– 2 Distributive Property y– Simplify. Solve the system of equations and find where the altitudes meet. 1 2 y= x+ 5 2 3 2 y=– x+ 1 x 2 33 2 5 3 33 2 2 2 5 33 = −2x + 2 2 + = x+ −14 = −2x 7=x 1 2 5 2 1 2 5 2 2 3 ���� has a slope of , then the altitude has a slope If 𝐴𝐴 3 3 = – (x – 9) 2 3 27 3=– x+ 2 2 3 33 y=– x+ 2 2 Point-slope form 3 m = – , (𝑥1 , 𝑦1 ) = C(9, 3) 2 Distributive Property Simplify. Original equations 1 5 Substitute x + for y. 1 2 2 Subtract x from each side. Subtract 2 33 2 from each side. Divide each side by –2. 7 2 5 2 y = x + = (7) + = + = 6 The coordinates of the orthocenter of △ABC are (7, 6). Exercises: COORDINATE GEOMETRY Find the coordinates of the orthocenter of the triangle with the given vertices. 2. S(1, 0), T(4, 7), U(8, −3) 1. J(1, 0), H(6, 0), I(3, 6) Chapter 5 12 Glencoe Geometry