Understanding Triangle Medians and Centroids for Geometry Problems
Explore the concept of triangle medians, centroids, and related theorems in geometry through examples and explanations. Learn how to find centroids of triangles with given vertices and solve centroid-related problems efficiently.
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Presentation Transcript
WARM UP March 11, 2014 Solve for x Solve for y 1. 3x x 2. (40 + y) 28
Medians of a Triangle The MEDIANS of a triangle join the vertex of one angle to the opposite side s midpoint. Every Triangle has 3 Medians.
The intersection of the medians is called the CENTROID.
Centroid Theorem The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint. 2x x
C How much is CW? CW = 2(WF) D CW = 2(13) E W 13 B A F CW = 26
C How much is WD? AW = 2(WD) 18 = 2(WD) D E W 18 B A F 9 = WD
How do you find the Centroid Given 3 points? Remember the midpoint formula ?1+ ?2 2 , 2 ) ?1+ ?2 ( The Centroid Formula is very similar. ?1+ ?2+ ?3 3 , 3 ?1+ ?2+ ?3 ) (
Example Find the centroid of a triangle whose vertices are (-1, -3), (2, 1) and (8, -4).
You Try!! Find the centroid of a triangle whose vertices are A(4, -1), B(2, 6), and C(9, -5).
YOU TRY!!!! In ABC, AN, BP, and CM are medians. If EN = 12, find AN. C N AE = 2(12)=24 P E B AN = AE + EN AN = 24 + 12 M A AN = 36
Mid-Segment of a Triangle The MID-SEGMENT of a triangle is a segment that joins two midpoints of two sides of a triangle.
The mid-segment of a triangle joins the midpoints of two sides of a triangle such that its length is half the length of the third side of the triangle.
Triangle Proportionality Theorem If a line is parallel to one side of the triangle and it intersects the other two sides, then the line divides the other two sides proportionally.
Examples Solve for x. 26= 13= 9 = 3x 3 = x ? 9 ? 9
YOU TRY!! Solve for x.
