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Learn about altitudes, bisectors, medians, and points of concurrency in triangles. Discover unique properties like the orthocenter, incenter, circumcenter, and centroid. Master the balance and equidistance concepts.
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altitudes bisectors medians SECTION 5-1 spi.3.2.J Jim Smith JCHS
There are 3 of each of these special segments in a triangle. The 3 segments are concurrent. They intersect at the same point. This point is called the point of concurrency. The points have special names and special properties.
Altitudes Start at a vertex and form a 90° angle with the line containing the opposite side. Orthocenter
The orthocenter can be located in the triangle, on the triangle or outside the triangle. Obtuse Right Legs are altitudes
Angle Bisectors Angle bisectors start at a vertex and bisect the angle. Incenter
Any point on an angle bisector is equidistance from the sides of the angle
This makes the Incenter an equidistance from all 3 sides
Perpendicular Bisectors A Perpendicular bisector of a side does not have to start at a vertex. It will form a 90° anglesand bisect the side. Circumcenter
Any point on the perpendicular bisector of a segment is equidistance from the endpoints of the segment. AB is the perpendicular bisector of CD A C D B
This makes the Circumcenter an equidistance from the 3 vertices
Median The median starts at a vertex and ends at the midpoint of the opposite side. Centroid
The centroid is 2/3’s of the distance from the vertex to the side. 32 10 2x 5 x X 16
The centroid is the center of balance for the triangle. You can balance a triangle on the tip of your pencil if you place the tip on the centroid
Quick notes Altitude .. Vertex .. 90° .. Orthocenter Angle Bisector.. Angle into 2 equal angles .. Incenter Perpendicular Bisector… 90° .. bisects side .. Circumcenter Median .. Vertex .. Midpoint of side ..Centroid
Give the best name for AB A A A A A B B B B B | | | | | | MedianAltitudeNone Angle Perp Bisector Bisector
Survival Training You’re Stranded On A Triangular Shaped Island. The Rescue Ship Can Only Dock On One Side Of The Island But You Don’t Know Which Side. At Which Point Of Concurrency Would You Set Up Camp So You Are An EqualDistance From All 3 Sides? INCENTER
What If The Ship Could Only Dock At One Of The Vertices? Would You Change The Location Of Your Camp ? If So, Where? YES CIRCUMCENTER
Where would you place a fire hydrant to make it equidistance to the houses and equidistance to the streets? ELM POST
Angle bisector for the streets Perpendicular bisector for houses ELM POST