University:
College Entrance Preparation
Course:
ACT Preparation (Mathematics)
Academic year:

2024
Views:

490

Pages:

12
Author:

Hundebuchesn

ANALYZING GEOMETRIC FIGURES USING ALGEBRAIC METHODS NAME: Key! ☺ PERIOD: Each set of vertices, when graphed, will create a polygon. Your job is to PROVE which type of polygon is represented. To do this, you must use the distance formula to show that certain sides are congruent, and the information you know about slope to prove which sides are parallel, perpendicular, or neither. You should use the grids to show the figure, but you can't use what it LOOKS like in order to prove anything! **Example:** A(-1,2) B(2,-3) C(-2,-2) Triangle AB = √((-3-2)² + (2+1)²) = √25 + 9 = √34 BC = √((-2-2)² + (-2+3)²) = √16 + 1 = √17 AC = √((-1+2)² + (2+2)²) = √1 + 16 = √17 Because AC = BC, △ABC is isosceles. mAB = (-3-2)/(2+1) = -5/3 mBC = (-2+3)/(-2-2) = 1/-4 mAB * mBC = -1 ⇒ perpendicular! Isosceles right triangle! 1. A(1,1) B(2,5) C(5,1) D(2,-3) AB = √((2-1)² + (5-1)²) = √1 + 16 = √17 BC = √((5-2)² + (1-5)²) = √9 + 16 = √25 = 5 CD = √((2-5)² + (-3-1)²) = √9 + 16 = √25 = 5 AD = √((1-2)² + (1+3)²) = √1 + 16 = √17 All sides equal! mAB = (5-1)/(2-1) = 4/1 = 4 mBC = (1-5)/(2-5) = -4/-3 = 4/3 mCD = (-3-1)/(2-5) = -4/-3 = 4/3 mAD = (-2-1)/(2-1) = -3/1 = -3 Opposite sides are parallel. All sides equal. Parallelogram (with all equal sides) Rhombus 2. A(2,-4) B(5,1) C(4,4) D(1,-1) AB = √((5-2)² + (1+4)²) = √9 + 25 = √34 BC = √((4-5)² + (4-1)²) = √1 + 9 = √10 CD = √((1-4)² + (-1-4)²) = √9 + 25 = √34 AD = √((2-1)² + (-4+1)²) = √1 + 9 = √10 Opposite sides are equal. mAB = (1+4)/(5-2) = 5/3 mBC = (4-1)/(4-5) = 3/-1 = -3 mCD = (-1-4)/(1-4) = 5/3 mAD = (-1+4)/(1-2) = 3/-1 = -3 4 right angles! Rectangle 1. A(1,1) B(5,2) C(-2,-3) AB = √((5-1)² + (2-1)²) = √16 + 1 = √17 BC = √((-2-5)² + (-3-2)²) = √49 + 25 = √74 AC = √((-2-1)² + (-3-1)²) = √9 + 16 = √25 = 5 mAB = (2-1)/(5-1) = 1/4 mBC = (-3-2)/(-2-5) = 5/7 mAC = (-3-1)/(-2-1) = 4/3 Isosceles right triangle [Image of the triangle ABC] 2. A(0,4) B(5,1) C(-2,-3) D(-1,2) AB = √((5-0)² + (1-4)²) = √25 + 9 = √34 BC = √((-2-5)² + (-3-1)²) = √49 + 16 = √65 CD = √((-1+2)² + (2+3)²) = √1 + 25 = √26 AD = √((0+1)² + (4+2)²) = √1 + 36 = √37 mAB = (1-4)/(5-0) = -3/5 mBC = (-3-1)/(-2-5) = 4/7 mCD = (-2+3)/(-1+2) = 1 mAD = (-1-4)/(1-0) = -5 (opposite sides parallel) Parallelogram [Image of the parallelogram ABCD] 3. A(1,4) B(1,4) C(2,2) D(2,-2) AB = √((1-1)² + (4-4)²) = 0 BC = √((2-1)² + (2-4)²) = √1 + 4 = √5 CD = √((2-2)² + (-2-2)²) = √16 = 4 AD = √((1-2)² + (4+2)²) = √1 + 36 = √37 No sides congruent Trapezoid [Image of the trapezoid ABCD] 4. A(3,1) B(4,2) C(2,3) AB = √((4-3)² + (2-1)²) = √1 + 1 = √2 BC = √((2-4)² + (3-2)²) = √4 + 1 = √5 AC = √((2-3)² + (3-1)²) = √1 + 4 = √5 Isosceles right triangle [Image of the triangle ABC] 5. A(2,2) B(5,1) C(-1,-4) D(-4,1) AB = √((5-2)² + (1-2)²) = √9 + 1 = √10 BC = √((-1-5)² + (-4-1)²) = √36 + 25 = √61 CD = √((-4+1)² + (1+4)²) = √9 + 25 = √34 AD = √((2+4)² + (2+1)²) = √36 + 9 = √45 mAB = (1-2)/(5-2) = -1/3 mBC = (-4-1)/(-1-5) = 5/6 mCD = (1+4)/(-4+1) = -5/3 mAD = (2+1)/(2+4) = 1/2 Parallel Parallel Perpendicular Perpendicular Square! [Image of the square ABCD]