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Chapter 5 Section 5.4

Chapter 5 Section 5.4. Medians and Altitudes. U SING M EDIANS OF A T RIANGLE. A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. D. U SING M EDIANS OF A T RIANGLE. centroid. centroid. centroid. acute triangle.

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Chapter 5 Section 5.4

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  1. Chapter 5Section 5.4 Medians and Altitudes

  2. USING MEDIANS OF A TRIANGLE A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. D

  3. USING MEDIANS OF A TRIANGLE centroid centroid centroid acute triangle right triangle obtuse triangle The three medians of a triangle are concurrent. The point of concurrency is called the centroid of the triangle. The centroid is always inside the triangle. The medians of a triangle have a special concurrency property.

  4. USING MEDIANS OF A TRIANGLE THEOREM AP= AD 2 2 2 BP = BF 3 3 3 CP= CE THEOREM 5.8 Concurrency of Medians of a Triangle The medians of a triangle intersect at a point that is two-thirdsof the distance from each vertex to the midpoint of the opposite side. If P is the centroid of ABC, then P

  5. USING MEDIANS OF A TRIANGLE The centroid of a triangle can be used as its balancing point. A triangular model of uniformthickness and density willbalance at the centroid of the triangle.

  6. Using the Centroid of a Triangle P is the centroid of QRS shown below and PT = 5. Find RT and RP. Because P is the centroid, RP = RT. 2 3 1 Then PT = RT – RP = RT 3 1 Substituting 5 for PT, 5 = RT, so RT = 15. 3 2 2 Then RP = RT = (15) = 10. 3 3 SOLUTION So, RP = 10 and RT= 15.

  7. Finding the Centroid of a Triangle Find the coordinates of the centroid of JKL. J (7, 10) N (3, 6) P L Choosethe median KN. Findthe coordinates of N, the midpoint of JL. (5, 2) The coordinates of N are K 3 + 7 6 + 10 , = 2 2 10 16 , = (5, 8) 2 2 (5, 8) SOLUTION The centroid is two thirds of the distance from each vertex to the midpoint of the opposite side.

  8. Finding the Centroid of a Triangle Find the coordinates of the centroid of JKL. J (7, 10) (3, 6) (5, 6) L Determine the coordinates ofthe centroid, which is • 6, or 4 units up from vertex K along the median KN. (5, 2) 2 3 K (5, 8) SOLUTION Findthe distance from vertex Kto midpoint N. The distance fromK (5,2) to N(5,8) is 8 – 2, or 6 units. N P M The coordinates of the centroid P are (5, 2 + 4), or (5, 6)

  9. USING ALTITUDES OF A TRIANGLE An altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side. An altitude can lie inside, on, or outside the triangle. Every triangle has three altitudes. The lines containing the altitudes are concurrent and intersect at a point called the orthocenterof the triangle.

  10. Drawing Altitudes and Orthocenters Where is the orthocenter of an acute triangle? SOLUTION Draw an example. The three altitudes intersect at G, a point inside the triangle.

  11. Drawing Altitudes and Orthocenters Where is the orthocenter of a right triangle? The two legs, LM and KM, are also altitudes. They intersect at the triangle’s right angle. SOLUTION This implies that the orthocenter is on the triangle at M, the vertex of the right angle of the triangle.

  12. Drawing Altitudes and Orthocenters Where is the orthocenter of an obtuse triangle? SOLUTION The three lines that contain the altitudes intersect at W, a point outside the triangle.

  13. THEOREM If AE, BF, and CD are the altitudes of ABC, then the lines AE, BF, and CD intersect at some point H. USING ALTITUDES OF A TRIANGLE THEOREM 5.9 Concurrency of Altitudes of a Triangle The lines containing the altitudes of a triangle are concurrent.

  14. Median Angle Bisector Perpendicular Bisector Altitude All Four

  15. PO = 11

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