Median of a triangle The median of a

Median of a triangle The median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side Every triangle has 1 median for each vertex, or 3 medians total

Centroid of a triangle • • • The centroid of a triangle is the point of concurrency of the 3 medians of a triangle. This point is also called the center of gravity When 3 or more lines intersect at 1 point, they are called concurrent lines

Centroid theorem 32 -1 • • The centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side CP = 2/3 CN BP = 2/3 BM AP = 2/3 AL

Using the centroid to find segment lengths • • • LA = 12, PN = 3. 1 Find AP AP =2/3 AL AP = 2/3(12) AP = 8 Find NC CP = 2/3 CN CP + PN = CN 2/3 CN + 3. 1 = CN 3. 1 = 1/3 CN 9. 3 = CN

Finding the centroid on a coordinate plane • • • Find the centroid of tri DEF with vertices D(3, 5), E(-2, 1), and F(-7, 3) Graph the triangle Find the midpoint of each segment midp DF = (-5, 4), midp DE = (-2. 5, 3) Since all 3 medians meet at the same point, the intersection of any 2 will give the location of the centroid.

• • • Find the equation of the line from point E to (-5, 4) or from (-2, 1) and (-5, 4) Slope = 3/-3 = -1 Y = mx+b so 1 = -1(-2) + b -1 = b y=-x-1 Find the equation of the line from point F to (-2. 5, 3) or from (-7, 3) and (-2. 5, 3) Slope is 0 so 3 = -x-1 4 = -x -4 = x so centroid is (-4, 3)

Find the centroid of tri RTS with R(-2, 2), T(2, 2), S(1, -2) • • • Find the medians Find the equations of 2 lines Find the coordinate of the centroid

Altitude of a triangle • The altitude of a triangle is the perpendicular segment from the vertex to the line containing the opposite side. • The orthocenter is the point of concurrency of the 3 altitudes of a triangle

Locating the orthocenter • • • Draw an acute, a right and an obtuse triangle Sketch the altitudes and find the orthocenter for each Is the orthocenter always in the interior of a triangle?

Use graph paper • Sketch the orthocenter of the triangle formed by (4, 2), (-1, 1) and (1, 6)