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### Understanding Triangle Centers: Concepts and Calculations in Geometry ###

Dive into the key concepts of triangle geometry, focusing on points such as circumcenters, incenters, centroids, and orthocenters. This guide explores the relationships between points and their distances, including perpendicular bisectors and medians, and provides insight into how to calculate lengths using the centroid properties. Learn about the significance of equidistance in triangle construction and the conditions under which specific points of concurrency exist. Hone your skills in solving problems involving triangle centers through practical examples and the application of the Pythagorean theorem. ###

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### Understanding Triangle Centers: Concepts and Calculations in Geometry ###

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Presentation Transcript

  1. GeometryGrab your clicker and get ready for the warm-up

  2. The distance from a point to a line can be called the “ ” distance • Parallel • Vertical • Perpendicular • Circumcenter • Bisector

  3. A point on a perpendicular bisector is from the two endpoints of the bisected segment • Equidistant • Perpendicular • Corresponding • Centroid • Midpoint

  4. A point on an angular bisector is equidistant from the two of the angle • Angles • Vertices • Right Angles • Sides • Incenters

  5. The point of concurrency for the perpendicular bisectors of a triangle is called the • Incenter • Orthocenter • Midpoint • Circumcenter • Centroid • Midsegment

  6. The point of concurrency for the angular bisectors of a triangle is called the • Incenter • Orthocenter • Midpoint • Circumcenter • Centroid • Midsegment

  7. A median of a triangle goes from the vertex to the of the opposite side • Circumcenter • Angle • Perpendicular • Centroid • Side • Midpoint • Orthocenter

  8. The point of concurrency for the medians of a triangle is called the • Incenter • Orthocenter • Midpoint • Circumcenter • Centroid • Midsegment

  9. An altitude goes from a vertex and is to the opposite side • Circumcenter • Angle • Perpendicular • Centroid • Side • Midpoint • Orthocenter

  10. The point of concurrency for the altitudes of a triangle is called the • Incenter • Orthocenter • Midpoint • Circumcenter • Centroid • Midsegment

  11. The circumcenter of a triangle is equidistant from the • Vertices • Incenter • Centroid • Perpendicular • Sides

  12. The incenterof a triangle is equidistant from the • Vertices • Incenter • Centroid • Perpendicular • Sides

  13. The Pythagorean Theorem for this right triangle would state: • a2 + b2 = c2 • f2 + g2 + h2 = 180 • f2 + g2 = h2 • h2 + g2 = f • g2 + h2 = 90 • g2 + h2 = f2 • g2 – h2 = f2

  14. Given C is the centroidand that XC = 8, determine CK • 16 • 14 • 12 • 10 • 8 • 6 • 4 • 2 • 1

  15. Given C is the centroidand that CZ = 3, determine CJ • 9 • 3 • 6 • 1.5 • 4.5 • Not possible • None of the above

  16. Given C is the centroidand that YI = 15, determine YC • 9 • 12 • 3 • 6 • 1.5 • 4.5 • 7.5 • 8 • Not possible • None of the above

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