Concurrent Lines, Medians, and Altitudes in Geometry
In geometry, concurrent lines intersect at a single point called the point of concurrency. Theorems state that perpendicular bisectors of triangle sides and angle bisectors are concurrent. The circumcenter, incenter, and centroid are points of concurrency with specific properties. Medians and altitudes also play crucial roles in triangle properties. Explore the concepts of concurrency in geometry with illustrations and examples.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
5 5- -3 CONCURRENT 3 CONCURRENT LINES, MEDIANS, AND LINES, MEDIANS, AND ALTITUDES ALTITUDES GEOMETRY GEOMETRY
Concurrent Concurrent: :When 3 or more lines intersect at one point. Point of Concurrency Point of Concurrency: : The point at which the three lines intersect.
Theorem 5 Theorem 5- -7: 7: The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. Theorem 5 Theorem 5- -8 8: The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.
Circumcenter of the Triangle Circumcenter of the Triangle: The point of concurrency of the perpendicular bisectors of a triangle The circle is circumscribed about circumscribed about the triangle
Ex. 1) Ex. 1) Find the center of the circle that you can Find the center of the circle that you can circumscribe about circumscribe about ???. .
Incenter of the Triangle Incenter of the Triangle: The point of concurrency of the angle bisectors of a triangle The circle is inscribed in inscribed in the triangle
Median: Median: a segment whose endpoints are a vertex and the midpoint of the opposite side
Theorem 5 Theorem 5- -8 8: The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. Centroid Centroid: Point of concurrency of the medians.
Ex. 2) Ex. 2)
Altitude: Altitude: the perpendicular segment from the vertex to the line containing the opposite side. The lines containing the altitudes of a triangle are concurrent at the ORTHOCENTER OF THE TRIANGLE Theorem 5 Theorem 5- -9 9: The lines that contain the altitudes of a triangle are concurrent. Acute Triangles Acute Triangles Right Triangles Right Triangles Obtuse Triangles Obtuse Triangles
Ex. 3) Ex. 3) D is the Centroid.
Ex. 4) Ex. 4) D is the Centroid.
