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Special Segments of Triangles: Definitions & Theorems

Learn about medians, centroids, altitudes, and orthocenters of triangles with step-by-step constructions, definitions, and theorems. Explore how these special segments intersect within triangles.

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Special Segments of Triangles: Definitions & Theorems

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  1. Medians and Altitudes of Triangles(Special Segments)

  2. Definitions and Theorems….. A segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. • Median-- • Centroid-- The point where the three medians of a triangle intersect.

  3. Definitions and Theorems….. The centroid is located the distance from the vertex to the midpoint on the opposite side. • Centroid Theorem-- Y B Centroid A D AZ = of AZ ZD = of AZ Z X C AD = of AZ

  4. Centroid Example BG = 8 GD = BD = EG = 2.4 GC = EC = AF = 9 AG = GF = Using Triangle ABC, find the segment lengths of AG and CE. B E F G A D C

  5. Construct three Medians Q (0,8) Centriod (2½, 4) R (6,4) P (2,0)

  6. Definitions and Theorems….. Is a perpendicular segment from the vertex to the opposite side of the triangle. • Altitude-- • Orthocenter-- The point where the three altitudes of a triangle intersect.

  7. Altitudes---height • Altitudes of a Triangle- • Orthocenter • The orthocenter can be located inside, outside or on the given triangle Y B A D Z X C

  8. Construct three altitudes Orthocenter (4.5, 3.5) Q (0,8) m = m = m = R (6,4) m = m = m = P (2,0)

  9. Steps for Constructing Special Segments: • 1. Slide 1 --- Steps for Constructing Perpendicular Bisectors • 2. Slide 2 --- Steps for Constructing Angle Bisectors • 3. Slide 3 --- Steps for Constructing medians and altitudes

  10. Perpendicular Bisectors • Graph the points • Find the midpoint of each side • Plot the midpoints • Find the slope of each side • Find the perpendicular slope of each side • From the midpoint, count using the perpendicular slope and plot another point • Draw a line segment connecting the midpoint and the point • The point where all 3 perpendicular bisectors cross is called the circumcenter You will need a straight edge for this construction

  11. Angle Bisectors • Graph the points and draw a triangle • Using a compass, draw an arc using one vertex as the center. This arc must pass through both sides of the angle • Plot points where the arc crosses the sides of the angle • From the points on the sides, create two more arcs (with the same radius) that cross. Plot a point where the two arcs cross • Draw a line segment from the vertex to the point where the small arcs cross. • When bisecting angles on a triangle, the point where all three angle bisectors cross is called the incenter You will need a compass and a straight edge for this construction

  12. Medians • Graph the points for the triangle • Find the midpoint of each side • Draw a line segment connecting the midpoint to the opposite vertex • The point where the 3 medians cross is called the centroid • Altitudes • Graph the points for the triangle • Find the slope of each side • Find the perpendicular slope of each side • From the opposite vertex, count using the perpendicular slope. Plot another point and draw a line segment to connect the vertex to this point • The point where all 3 altitudes cross is called the orthocenter You will need a straight edge for both of these constructions

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