Lesson 5 – 2 Medians and Altitudes of Triangles Geometry Lesson 5 – 2 Medians and Altitudes of Triangles Objective: Identify and use medians in triangles. Identify and use attitudes in triangles.

Median Median of a triangle A segment with endpoints at a vertex of a triangle and the midpoint of the opposite side.

Centroid Centroid Centroid Theorem The point of concurrency of the medians of a triangle. Centroid Theorem The medians of a triangle intersect at a point called the centroid that is two thirds of the distance from each vertex to the midpoint of the opposite side.

Centroid… PK = 5 Find AP 10 x 2x BP = 12 Find PL. 6 JC = 15 PK + AP = AK JC = 15 Find JP. 5 PK + 2(PK) = AK

In triangle ABC, Q is the centroid and BE = 9 Find BQ Find QE BQ = 2(QE) = 3 OR 6 = 2(QE) 3 = QE

In triangle ABC, Q is the centroid and FC = 14 Find FQ Find QC = 5 QC = 2(FQ) QC = 10 QC = 2(5)

In triangle JKL, PT = 2. Find KP. How do you know that P is the centroid? KP = 2(PT) = 2(2) = 4 OR KP = 4

In triangle JKL, RP = 3.5 and JP = 9 Find PL Find PS PL = 2(RP) = 2(3.5) = 7 PS = 4.5 JP = 2(PS) 9 = 2(PS)

A performance artist plans to balance triangular pieces of metal during her next act. When one such triangle is placed on the coordinate plane, its vertices are located at (1, 10) (5, 0) and (9,5). What are the coordinates of the point where the artist should support the triangle so that it will balance. The balance point of a triangle is the centroid.

Find the midpoint of the side(s) that could make a Graph the points. Hint: To make it easier look for a vertical or horizontal line between a midpoint of a side and vertex. Find the midpoint of the side(s) that could make a vertical or horizontal line. Find the midpoint of AB. Midpoint of AB = = (3, 5) Let P be the Centroid, where would it be? From the vertex to the centroid is 2/3 of the whole.

Count over from C 4 units and that is P Centroid: (5, 5)

A second triangle has vertices (0,4), (6, 11. 5), and (12,1) A second triangle has vertices (0,4), (6, 11.5), and (12,1). What are the coordinates of the point where the artist should support the triangle so that it will balance? Explain your reasoning. Centroid: (6, 5.5)

Altitude Altitude of a triangle Draw a right triangle and identify A perpendicular segment from a vertex to the side opposite that vertex. Draw a right triangle and identify all the altitudes.

Orthocenter Orthocenter The lines containing the altitudes of a triangle are concurrent, intersecting at a point called the orthocenter.

Find the orthocenter The vertices of triangle FGH are F(-2, 4), G(4,4), and H(1, -2). Find the coordinates of the orthocenter of triangle FGH. Graph the points. Cont…

Find an equation from F to GH. Slope of GH. m = 2 New equation is perpendicular to segment GH. Point F (-2, 4) m = -1/2 y = mx + b 3 = b Cont…

Find an equation from G to FH. Slope of segment FH m = -2 New equation is perpendicular to segment FH. Point G (4, 4) m = 1/2 2 = b Cont…

The orthocenter can be found at the intersection of our 2 new equations. How can we find the orthocenter? If the orthocenter lies on an exact point of the graph use the graph to name. If it does not lie on a point use systems of equations to find the orthocenter. System of equations: Cont…

By substitution. Orthocenter (1, 2.5) 1 = x y = 2.5

Summary Perpendicular bisector

Summary Angle bisector

Summary Median

Summary Altitude

Homework Pg. 337 1 – 10 all, 12 - 20 E, 27 – 30 all, 48 – 54 E