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# Coordinate Geometry Basics

Created by
@SimplerSunflower

### What is the correct formula to calculate the distance between two points with coordinates $$(x_1, y_1)$$ and $$(x_2, y_2)$$?

• d = (x_2 - x_1) + (y_2 - y_1)
• d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} (correct)
• d = \\sqrt{(x_2 + x_1)^2 + (y_2 + y_1)^2}
• d = (x_2 + x_1) - (y_2 - y_1)
• ### What is the slope of a line that is perfectly horizontal?

• Negative
• Zero (correct)
• Positive
• Undefined
• ### Which standard form equation represents a circle?

• \$$y = mx + b\$$
• \$$(x - h)^2 + (y - k)^2 = r^2\$$ (correct)
• \$$Ax + By = C\$$
• \$$rac{(x - h)^2}{a^2} + rac{(y - k)^2}{b^2} = 1\$$
• ### In the point-slope form of a line, which variables are represented by $$(x_1, y_1)$$?

<p>Any point on the line</p> Signup and view all the answers

### If two lines are perpendicular, what relation holds true about their slopes?

<p>They are negative reciprocals of each other</p> Signup and view all the answers

### Which of the following equations represents a parabola?

<p>$$y = ax^2 + bx + c$$</p> Signup and view all the answers

### What characteristics do parallel lines share?

<p>Identical slopes and never intersect</p> Signup and view all the answers

### What represents the midpoint between two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$?

<p>$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\\right)$$</p> Signup and view all the answers

## Study Notes

### Coordinate Geometry

• Definition: A branch of mathematics that studies geometric figures using a coordinate system.

• Coordinate System:

• Cartesian Coordinates: Uses a pair of numbers (x, y) to represent points on a plane.
• Origin: The point (0, 0).
• Axes:
• x-axis (horizontal)
• y-axis (vertical)
• Distance Formula:
• Distance between two points ((x_1, y_1)) and ((x_2, y_2)):
• (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2})
• Midpoint Formula:

• The midpoint M of a segment connecting points ((x_1, y_1)) and ((x_2, y_2)):
• (M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right))
• Slope of a Line:

• The slope (m) between two points:
• (m = \frac{y_2 - y_1}{x_2 - x_1})
• Identifies line steepness and direction:
• Positive slope: Rising line
• Negative slope: Falling line
• Zero slope: Horizontal line
• Undefined slope: Vertical line
• Equation of a Line:

• Slope-intercept form:
• (y = mx + b) where (m) is slope and (b) is y-intercept.
• Point-slope form:
• (y - y_1 = m(x - x_1))
• Standard form:
• (Ax + By = C)
• Types of Lines:

• Parallel Lines: Same slope, never intersect.
• Perpendicular Lines: Slopes are negative reciprocals ((m_1 \cdot m_2 = -1)).
• Conic Sections:

• Circle:
• Standard equation: ((x - h)^2 + (y - k)^2 = r^2) where ((h, k)) is the center and (r) is the radius.
• Ellipse:
• Standard equation: (\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1)
• Parabola:
• Standard equation: (y = ax^2 + bx + c)
• Hyperbola:
• Standard equation: (\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1)
• Applications:

• Used in physics, engineering, computer graphics, and navigation.
• Tools:

• Graphing calculators and software for visual representation.
• Coordinate grids for plotting and analyzing geometric properties.

### Coordinate Geometry

• A mathematical discipline that interprets geometric figures through a coordinate framework.

### Coordinate System

• Cartesian Coordinates: Employs (x, y) pairs to indicate positions on a plane.
• Origin: Located at (0, 0), serving as the reference point.
• Axes:
• x-axis: horizontal line
• y-axis: vertical line

### Distance Formula

• To compute the distance (d) between two coordinates ((x_1, y_1)) and ((x_2, y_2)):
• Formula: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2})

### Midpoint Formula

• Calculates the midpoint (M) of a line segment connecting ((x_1, y_1)) and ((x_2, y_2)):
• Formula: (M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right))

### Slope of a Line

• The slope (m) indicates the angle of inclination between two coordinates:
• Formula: (m = \frac{y_2 - y_1}{x_2 - x_1})
• Interpretations:
• Positive slope: rising line
• Negative slope: falling line
• Zero slope: horizontal line
• Undefined slope: vertical line

### Equation of a Line

• Slope-intercept form: (y = mx + b) with (m) as slope and (b) as y-intercept.
• Point-slope form: (y - y_1 = m(x - x_1)).
• Standard form: (Ax + By = C) where (A), (B), and (C) are constants.

### Types of Lines

• Parallel Lines: Lines sharing the same slope but never meet.
• Perpendicular Lines: Lines that intersect at right angles; their slopes are negative reciprocals ((m_1 \cdot m_2 = -1)).

### Conic Sections

• Circle: Standard equation ((x - h)^2 + (y - k)^2 = r^2) identifies center ((h, k)) and radius (r).
• Ellipse: Given by (\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1), describes an elongated circle.
• Parabola: Standard form (y = ax^2 + bx + c) depicts a U-shaped curve.
• Hyperbola: Standard equation (\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1) forms two separate curves.

### Applications

• Integral in fields such as physics, engineering, computer graphics, and navigation.

### Tools

• Utilize graphing calculators and software for visualizing data.
• Coordinate grids are essential for mapping and exploring geometric relationships.

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## Description

Explore the fundamental concepts of coordinate geometry, including Cartesian coordinates, the distance and midpoint formulas, and the slope of a line. This quiz will test your understanding of how to apply these essential mathematical principles to solve various problems. Perfect for students looking to solidify their grasp on the topic.

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