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Questions and Answers
What are the coordinates of two points?
What are the coordinates of two points?
What is the formula for slope?
What is the formula for slope?
m = (y₂ - y₁) / (x₂ - x₁)
What is the Midpoint Formula?
What is the Midpoint Formula?
M(x, y) = ((x₁ + x₂)/2, (y₁ + y₂)/2)
What is the Distance Formula?
What is the Distance Formula?
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What is the equation of a line in y-intercept form?
What is the equation of a line in y-intercept form?
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What is the Point-Slope Form of a line?
What is the Point-Slope Form of a line?
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Parallel lines have the same slope.
Parallel lines have the same slope.
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For perpendicular lines, the slopes are opposite reciprocals.
For perpendicular lines, the slopes are opposite reciprocals.
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What is a parallelogram?
What is a parallelogram?
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What defines a rectangle?
What defines a rectangle?
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What defines a rhombus?
What defines a rhombus?
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What is the definition of a square?
What is the definition of a square?
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What is an isosceles trapezoid?
What is an isosceles trapezoid?
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How can you prove that a figure is a parallelogram?
How can you prove that a figure is a parallelogram?
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What is required to prove a figure is a rectangle?
What is required to prove a figure is a rectangle?
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What conditions must be met to prove a figure is a rhombus?
What conditions must be met to prove a figure is a rhombus?
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How can you prove a figure is a trapezoid?
How can you prove a figure is a trapezoid?
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What is the method to prove a figure is an isosceles trapezoid?
What is the method to prove a figure is an isosceles trapezoid?
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Study Notes
Basic Coordinate Concepts
- Points are defined as (x₁, y₁) and (x₂, y₂), where x and y are the coordinates in a two-dimensional space.
Slope
- The slope ( m ) indicates the steepness of a line, calculated as ( m = \text{rise/run} = \frac{y₂ - y₁}{x₂ - x₁} ).
Midpoint Formula
- The midpoint ( M(x, y) ) of two points is given by ( M = \left(\frac{x₁ + x₂}{2}, \frac{y₁ + y₂}{2}\right) ).
Distance Formula
- The distance ( d ) between two points can be calculated using the formula ( d = \sqrt{(y₂ - y₁)² + (x₂ - x₁)²} ).
Equations of Lines
- Includes various forms such as slope-intercept and point-slope forms.
y-intercept Form
- The linear equation is expressed as ( y = mx + b ), where ( m ) is the slope and ( b ) is the y-intercept.
Point-Slope Form
- The equation takes the form ( y - y₁ = m(x - x₁) ), useful for lines through a specific point with a known slope.
Parallel Lines
- Two lines are parallel if they have the same slope, indicating they will never intersect.
Perpendicular Lines
- Lines are perpendicular if their slopes are opposite reciprocals, meaning the product of their slopes equals -1.
Partition Segments
- The partition ratio divides a line segment into specified parts based on given ratios.
Partition Ratio
- A partition ratio like 2:3 compares segments AB to BC relative to total length AC, represented as ( \frac{2}{3} ).
AB Distance from A Ratio
- The ratio of distance from point A to segment AB relative to total length AC can be expressed as ( \frac{2}{2 + 3} = \frac{2}{5} ).
Properties of Quadrilaterals
- Quadrilaterals include various shapes such as parallelograms, rectangles, rhombuses, squares, and trapezoids, each having unique properties.
Parallelogram
- Defined as a quadrilateral with two pairs of parallel sides.
- Properties: Opposite sides are congruent and parallel; opposite angles are congruent; diagonals bisect each other.
Rectangle
- A special type of parallelogram characterized by having four right angles.
- Inherits all properties of a parallelogram, plus diagonals are congruent.
Rhombus
- A parallelogram with four congruent sides.
- Shares properties of a parallelogram, with diagonals being perpendicular and bisecting opposite angles.
Square
- A parallelogram with four congruent sides and four right angles.
- Incorporates all properties of parallelograms, plus the diagonals are perpendicular and bisect opposite angles.
Trapezoid
- Defined as a quadrilateral with exactly one pair of parallel sides, distinguishing it from other quadrilaterals.
Isosceles Trapezoid
- A trapezoid with congruent base angles.
- Properties include congruent diagonals, supplementary opposite angles, and a median parallel to the bases, with a length that is half the sum of the parallel sides.
Proving Quadrilaterals
- Different methods exist to establish the properties and classifications of figures as quadrilaterals.
To Prove a Figure is a Parallelogram
- Show that either the diagonals bisect each other, both pairs of opposite sides are parallel, both pairs are congruent, or one pair is both congruent and parallel.
To Prove a Figure is a Rectangle
- Confirm it is a parallelogram via one of the four methods, and additionally verify it has either one right angle or that the diagonals are congruent.
To Prove a Figure is a Rhombus
- Confirm the figure is a parallelogram using the four methods and additionally show either the diagonals are perpendicular or two adjacent sides are congruent.
To Prove a Figure is a Square
- Establish the figure as a rectangle and confirm that two adjacent sides are congruent, or demonstrate it is a rhombus with one right angle.
To Prove a Figure is a Trapezoid
- Confirm the quadrilateral has only one pair of opposite sides that are parallel.
To Prove a Figure is an Isosceles Trapezoid
- Show it is a trapezoid and further validate that either the diagonals are congruent or the legs are congruent.
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