Geom6 3MediansAndAltitudesOfTrianglesNotes.notebook March 23, P Medians and Altitudes of Triangles

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1 Geometry P Medians and Altitudes of Triangles Review Circumcenter * The Perpendicular Bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. We call this point the Circumcenter because you can Circumscribe a circle about the triangle using this point as the center.

2 Review Incenter * The Angle Bisectors of a triangle are concurrent at point equidistant from the sides. We call this point the Incenter because you can Inscribe a circle inside the triangle using this point as the center. Review Circumscribed About Inscribed In The circle must touch all three vertices of the triangle. The center of the circle is the circumcenter. The circle must touch all three sides of the triangle. The center of the circle is the incenter.

3 Median: A segment from a vertex to the midpoint of the opposite side. Centroid of a Triangle: Point of Concurrency of the Medians of a Triangle. C Example. Point is the Centriod The Centroid is also called the center of gravity of a triangle because it is the point where a triangular shape will balance. Median Centroid To Find a Midpoint of a Segment... Use Perp Bisector to find the middle of each segment...

4 The distance from a vertex to the centroid is the distance from each vertex to the midpoint of the opposite side. Therefore, the remaining portion of the segment is of the length. In RST, point Q is the centroid, and SQ = 8. Find QW and SW. Altitude: The Perpendicular segment from a vertex to the opposite side. Orthocenter Point of Concurrency of the lines containing the Altitudes of a Triangle... Example. Point G is the Orthocenter.

5 ***Unlike angle bisectors and medians, an altitude can be a side or it may lie inside or outside the triangle. Altitude Orthocenter

6 Summary

7 Centroid Center of Gravity Use Medians

8 Orthocenter Use Altitudes

9 HW# WS Please put your name and class period at the top of the homework. Also include the homework number. Find the coordinates of the centroid of RST with vertices R(2, 1), S(5, 8), and T(8, 3).

10 Find the coordinates of the orthocenter of XYZ with vertices X( 5, 1), Y( 2, 4), and Z(3, 1). Where should a new library be built so that it is equidistant to all three towns? Three snack carts sell frozen yogurt from points A, B, and C outside a city. Each of the three carts is the same distance from the frozen yogurt distributor. Find the location of the distributor.

11 The Jacksons want to install the largest possible circular pool in their triangular backyard. Where would the largest possible pool be located? There are three paths through a triangular park. Each path goes from the midpoint of one edge to the opposite corner. The paths meet at point P. 1. Find PS and PC when SC = 2100 feet. 2. Find TC and BC when BT = 1000 feet. 3. Find PA and TA when PT = 800 feet.

12 A city wants to place a lamppost on the boulevard shown so that the lamppost is the same distance from all three streets. Should the location of the lamppost be at the circumcenter, incenter, or the centroid of the triangular boulevard? Explain.