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Geometry Chapter 5

Geometry Chapter 5. By: Benjamin Koch and Satya Nayagam. Section 5-1 Perpendicular and Angle Bisectors. Definitions Equidistant - a point that is the same distance from two or more objects Locus - set of points that satisfies a given condition. Section 5-1 Theorems.

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Geometry Chapter 5

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  1. Geometry Chapter 5 By: Benjamin Koch and Satya Nayagam

  2. Section 5-1Perpendicular and Angle Bisectors Definitions • Equidistant- a point that is the same distance from two or more objects • Locus- set of points that satisfies a given condition

  3. Section 5-1 Theorems 5-1-1: Perpendicular Bisector Theorem- If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment 5-1-2: Converse of Perpendicular Bisector Theorem- If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment 5-1-3: Angle Bisector Theorem- If a point is on the bisector of an angle, then it is equidistant from the sides of the angle 5-1-4: Converse of Angle Bisector Theorem- If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle

  4. Section 5-1 Example

  5. Section 5-2Bisectors of Triangle Definitions • Concurrent- 3 or more lines that intersect at one point • Point of Concurrency- point where concurrent lines intersect • Circumcenter- point of intersection of perpendicular bisectors of triangle, is equidistant from vertices of triangle

  6. Circumscribed- circle that contains all vertices of triangle • Incenter- the point of concurrency of the angle bisectors • Inscribed- circle that intersects each line at exactly on point

  7. Section 5-2 Theorems 5-2-1: Circumcenter Theorem- The circumcenter of a triangle is equidistant from the vertices of the triangle 5-2-2: Incenter Theorem- The incenter of a triangle is equidistant from the sides of the triangle

  8. Section 5-2 Example

  9. Section 5-3Medians and Altitudes of Triangles Definitions • Median of Triangle- segment whose endpoints are a vertex of triangle and the midpoint of opposite side • Centroid- always inside triangle, point where triangular region will balance, intersection of medians • Altitude of Triangle- perpendicular segment from vertex to line containing opposite side, triangle has 3, can be inside, outside, or on triangle • Orthocenter of a Triangle- the point of congruency of the altitudes

  10. Section 5-3 Theorem 5-3-1 Centroid Theorem- centroid is located 2/3 of the distance from each vertex to midpoint of opposite side

  11. Section 5-3 Example

  12. Section 5-4 The Triangle Midsegment Theorem Definitions • Midsegment of a Triangle- a segment that joins the midpoints of two sides of the triangle

  13. Section 5-4 Theorems 5-4-1 Triangle Midsegment Theorem- midsegment of triangle is parallel to a side of triangle, and its length is half the length of that side

  14. Section 5-4 Example

  15. Section 5-5 Indirect Proof and Inequalities in One Triangle Definitions • Indirect Proofs- begin by assuming conclusion is false, then show assumption leads to contradiction, called proof by contradiction

  16. Section 5-5 Theorems 5-5-1: Angle-Side Relationships in Triangles- If two sides of a triangle are not congruent, then the larger angle is opposite the longer side 5-5-2: Angle-Side Relationships in Triangles- If two angles of a triangle are not congruent, then the longer side is opposite the larger angle 5-5-3: Triangle Inequality Theorem- The sum of any two side lengths of a triangle is greater than the third side length

  17. Section 5-5 Example

  18. Section 5-6 Inequalities in Two Triangles Theorems 5-6-1 Hinge Theorem- if 2 sides of 1 triangle are congruent to 2 sides of another triangle and included angles aren’t congruent, then longer 3rd side is across from larger included angle, Converse: larger included angle is across from longer 3rd side.

  19. Section 5-6 Example

  20. Section 5-7 Pythagorean Theorem Definitions Definitions • Pythagorean Triple- three nonzero whole numbers that works out into the three sides of a right triangle

  21. Section 5-7 Pythagorean Theorem Theorems 5-7-1: Converse of Pythagorean Theorem-if sum of squares of lengths of 2 sides of triangle is equal to square of length of 3rd side, then triangle is right 5-7-2: Pythagorean Inequalities Theorem-if c^2 > a^2+b^2 then triangle abc is obtuse, if c^2 < a^2+b^2 then triangle abc is acute

  22. Section 5-7 Example

  23. Section 5-8 Applying Special Right Triangles Theorems 5-8-1: 45-45-90 degrees Triangle Theorem- In a 45-45-90 triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times the square root of 2 5-8-2: 30-60-90 degrees Triangle Theorem- In a 30-60-90 triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times the square root of 3

  24. Section 5-8 Example

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