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Geometry Theorems and Concurrency Points in Triangles

Explore Perpendiculars, Bisectors, Medians, and Altitudes in triangles, including Incenters, Circumcenters, and Orthocenters. Understand triangle inequalities and theorems.

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Geometry Theorems and Concurrency Points in Triangles

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  1. Chapter 5 Notes

  2. 5.1 – Perpendiculars and Bisectors

  3. A perpendicular bisector of a segment is a line or ray that is perpendicular to the segment at the midpoint. D A C B Perpendicular Bisector Theorem, If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. Converse of the Perpendicular Bisector Theorem, If a point is equidistant from the endpoints of a segment, then the point is on the perpendicular bisector of the segment.

  4. Distance from a point to a line is defined to be the length of the perpendicular segment from the point to the line (or plane) Which one represents distance?

  5. B C A D Angle Bisector Theorem, If a points lies on the bisector of an angle, then the point is equidistant from the sides of the angle. Converse of the Angle Bisector Theorem, If a point is equidistant from the sides of an angle, then the point lies on the angle bisector.

  6. Constructing a perpendicular to a line through a given point on the line. 1) From the given point, pick any arc and mark the circle left and right. 2) Those two marks are your endpoints, and construct a perpendicular bisector just like the previously. Justification. Line is perpendicular by construction, 3 is on the bisector because it is equidistant to both endpoints (because radii are equal), so the line is going through the point.

  7. Which segment is the perpendicular bisector, how do you know? Find DK. Find US. Find SK. Find CK What could DK be so that the segment would NOT be a perpendicular bisector, how would you know? U 8 5 D C 12 S K

  8. R lies on what? How do you know? OM is the angle bisector of EOT Find MT. G E 8 6 M O R T Y

  9. G a xo E 30o 8 6 T b yo O M R

  10. 5.2 – Bisectors of a Triangle5.3 – Medians and Altitudes of a Triangle

  11. Where multiple lines meet is called the point of concurrency. The lines that go through that point are called concurrent lines.

  12. The point of concurrency of angle bisectors is called an INCENTER Thrm: The angle bisectors of triangle intersect in a point that is equidistant from the three sides of a triangle. Justification, points on angle bisector are equidistant to the sides, then transitive.

  13. The point of concurrency of perpendicular bisectors is called a CIRCUMCENTER Thrm: Perpendicular bisectors of the sides of a triangle intersect in a point that is equidistant to all the vertices. Justification, points on perpendicular bisector are equidistant to the endpoints, then transitive. So to help keep track of things, it’s like they go with the other, angle bisectors equidistant to sides. Perpendicular bisectors equidistant to vertices.

  14. Inside or outside, where do the points of concurrency meet? Make a sketch and see CIRCUMCENTERS INCENTERS Acute – Inside Right – On side Obtuse – Outside All inside

  15. Red lines are angle bisectors. MA = -7x MB = x2 – 8 A M B

  16. Blue lines are perpendicular bisectors 3x - 4 14 5

  17. Median – A line from the midpoint to the vertex Where they all meet is the CENTROID The distance from the Centroid to the vertex is 2\3 the median. The distance from the Centroid to the midpoint is 1\3 the median.

  18. D M U G C K S 5 DU = UG = KS = 6 9 CU = GS = CS = US = DC = CK = DK = 18 CM = SM = GM = DS = DG = 6 24 GK = CG = DM =

  19. The point of concurrency of altitudes is called an ORTHOCENTER Thrm: Altitudes all meet at point. Nothing special about it.

  20. Inside or outside, where do the points of concurrency meet? Make a sketch and see Orthocenters Centroids Acute – Inside Right – On vertex Obtuse – Outside All inside

  21. 5.5 – Inequalities in One Triangle

  22. Terminology and Concepts Terminology  The side opposite the angle is the side that is across from and doesn’t touch the angle. Concept  The sides and angles opposite from each other often relate to each other. Angles will use an uppercase letter, and the side opposite will use a lower case letter or segment name. A c b B C a

  23. Theorem 5.10  If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side. R T S Theorem 5.11  If one angle of a triangle is larger than the 2nd angle, then the side opposite the first angle is longer than the side opposite the 2nd angle. Basically, big angle goes with big side, small angle goes with small side.

  24. Exterior angle inequality theorem: The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles. NONADJACENT means not attached to. R T S

  25. Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. AB + BC > AC AC + BC > AB AB + AC > BC A B C

  26. Pick the greater angle, 1 or 2? 8 11.1 11 9 2 Name the sides, shortest to longest. R ____ < ____ < ____ 50o T S

  27. Is it possible for a triangle to have these side lengths? Given two side lengths, find the possible lengths for the 3rd side ‘x’ 5, 6, 7 5, 6 10, 10, 10 2, 10 1, 1, 2 1, 9 1.1, 1.2, 1.3 4.9, 5, 10

  28. 5.6 – Indirect Proof and Inequalities in Two Triangles

  29. Hinge Theorem  If two sides one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. Fancy talk for two sides same, one angle bigger than other, then side is bigger D A E B C F

  30. Converse of Hinge Theorem  If two sides one triangle are congruent to two sides of another triangle, but the 3rd side of the first triangle is longer than the 3rd side of the second, then the included angle of the first triangle is larger than the included angle of the second. Fancy talk for two sides same, one sidee bigger than other, then angle is bigger D A E B C F

  31. Lots of examples of both types, along with algebra styles

  32. List the angles and sides in order S S U 2 U 35o 1 14 45o 30o D 70o C D 70o 13 C K K ____ < ____ ____ < ____ ____ < ____ < ____ ____ < ____ < ____ student

  33. Indirect Proof How to write an indirect proof 1. Assume temporarily that the conclusion is not true. 2. Reason logically until you reach a contradiction of the known fact. 3. Point out the temporary assumption is false, so the conclusion must be true.

  34. Practice  Write the untrue conclusion

  35. 1 a 3 b

  36. a 1 3 b

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