Advanced Math Extension: Analytic Geometry

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Presentation transcript:

Advanced Math Extension: Analytic Geometry 6-1 Advanced Math Extension: Analytic Geometry Summary of Methods Commonly Used in Coordinate Proofs: To prove line segments equal, use the distance formula to show that they have the same length. To prove non-vertical lines parallel, show that they have the same slope. To prove lines perpendicular, show that the product of their slopes is 1. To prove that two line segments bisect each other, use the midpoint formula to show that each segment has the same midpoint. To show that lines are concurrent, show that their equations have a common solution.

Coordinate Geometry uses actual points. Analytic Geometry uses variables; zero is the only acceptable number. You use definitions and formulas (midpoint, distance, and slope). (0, 0) x y (a, 0) (0, b) Right Triangle Place a geometric figure in a convenient position on the coordinate plane (usually placing a vertex on the origin). Scalene Triangle (0, 0) x y (b, c) (a, 0) (0, 0) x y (a, b) (2a, 0) Isosceles Triangle (a, 0) x y (0, b) (a, 0) Isosceles Triangle

Analytic Proofs Place a geometric figure in a convenient position on the coordinate plane (usually placing a vertex on the origin). (0, 0) x y (0, 0) x y Parallelogram Rectangle using midpoints (b, c) (a+b, c) (0, 2b) (2a, 2b) (a, 0) (2a, 0) (0, 0) x y (0, 0) x y Square Rectangle w/out midpoints (0, a) (0, b) (a, a) (a, b) (a, 0) (a, 0)

Trapezoid using midpoints Isosceles Trapezoid Trapezoid using midpoints (0, 0) x y y (-b, c) (b, c) Q R (2b, 2c) (2d, 2c) P S (2a, 0) (-a, 0) (a, 0) x

Trapezoid Midsegment Theorem y The midsegment of a trapezoid is parallel to the bases The length of the midsegment of a trapezoid is half the sum of the lengths of the bases. PROVE IT! (2b, 2c) (2d, 2c) x (0, 0) (2a, 0)

Analytic Proofs A3-Block B1 #6* Marissa Hayley #8 ChristianA JB #9 JoshC Sliker #1 Allie Sasha #2 Max #3 Jennie #4 Kevin Mikayla #5 Eric #6 Dena Rachel #7 Courtney Sarah #8 Michael Patrick #9 Chris Abby #1 Carolyn Gabby #2/3 ChristianP David #4 Jess Kayleigh #5* Alek Nathaniel JoshS Zoe #6* Tyler John

Analytic Proofs E-Block B4 #6* Nate Mallorie #8 Boo Kasey #9 Ben K Austin #1 Sam G Bobby #2/3 Kara Alicia #4 Abby Aleah #5* Deanna Kasi *Ask for hint on placement #6* Evan Alex #8 Jess Floriana #9 Annalee Sam T. #1 Kim Ben T. #2/3 Conner Connor #4 Richard Anna #5* Alan David #5* Rachel Natalie #6* Kevin Jack