About This Chapter
Understanding Analytical Geometry - Chapter Summary
Top instructors in this chapter simplify the process of understanding analytical geometry. Review fun and engaging lessons to boost your knowledge of the Cartesian coordinate system, learn how to write equations in standard form, and study measurements of angles and lengths and much more. Upon completion of this chapter, you will have the knowledge to:
- Plot points and graph lines using the Cartesian coordinate system
- Find the distance between two points as well as a point and a line
- List and describe properties of concurrent lines in a triangle
- Convert between two forms of equations for a circle
- Determine the measurements of angles and lengths involving tangents, chords and secants
- Solve real-world problems using parametric equations
- Write the equation of a parabola, ellipse and hyperbola in standard form
- Graph in three dimensions
Study analytical geometry your way by choosing lessons to review, navigating them in any order, and visiting them as often as you'd like until you gain full comprehension. All lessons are available around-the-clock and can be viewed using any computer or mobile device. Lesson quizzes and a chapter exam are available to assess your knowledge of analytical geometry. If you develop questions about specific lesson topics while studying this chapter, feel free to send them to our experts.
1. The Cartesian Coordinate System: Plotting Points & Graphing Lines
The Cartesian coordinate system establishes a specific position on a graph using equations based on the x and y axes. Learn how to use the Cartesian coordinate system, plot points on a graph, solve line equations using the table method or intercept method, and practice graphing lines.
2. How to Find the Distance Between Two Points
The distance between two points on a coordinate plane tells us how far apart the objects are from one another. Learn how to find this distance using the distance formula and some step-by-step examples.
3. How to Find the Distance Between a Point & a Line
Finding the distance from a point to a line is useful in many science and engineering applications. In this lesson, we calculate this distance using a formula, as well as with some fundamental concepts from geometry and algebra.
4. Properties of Concurrent Lines in a Triangle
Concurrent lines are those that meet in a single point when three or more are present. Explore the properties of concurrent lines in triangles through the concepts of the centroid, orthocenter, incenter, and circumcenter.
5. How to Find the Equation of a Circle
In mathematics, circles have two forms of equations. Learn how to find the equation of a circle. Recognize that circle equations have a standard form, as well as a general form, and explore several examples to gain an understanding of each form.
6. Measurements of Angles Involving Tangents, Chords & Secants
An angle refers to the space that is created when two lines intersect or meet. Learn how to calculate measurements about angles that are created by tangents, chords, and secants in a circle, including two chords, tangent and chord, two tangent lines, tangent and secant, and two secant lines.
7. Measurements of Lengths Involving Tangents, Chords and Secants
A circle has measurements of lengths that intersect at different points on the curve. Learn about intersecting chords, secants, and tangents and their relationship to each other on the circle.
8. Parametric Equations in Applied Contexts
Parametric equations are those that involve two or more variables, and are expressed by defining each variable in terms of only one other variable, called the parameter. See how these are found and used by solving several examples of parametric equations applied to real-life contexts.
9. How to Write the Equation of a Parabola in Standard Form
A parabola is an open curve that is highly distinguishable due to its symmetry along a certain axis. Learn how to write the equation of a parabola in standard form and understand the significance of the vertex and focus of a parabola.
10. How to Write the Equation of an Ellipse in Standard Form
The ellipse equation in standard form involves the location of the ellipse's center and its size. Learn what the standard form of an ellipse equation is, how to identity the center and size of the ellipse, and how to write the equation.
11. How to Write the Equation of a Hyperbola in Standard Form
The equation of a hyperbola can be written in standard form to provide useful information that can be observed without a graph. Learn how to use centers, vertices, and foci to write the equation of a hyperbola in standard form.
12. Graphing in Three Dimensions: Method & Formula
Unlike 2-D graphs, 3-D graphs help in understanding location relative to the three-dimensional world. Discover the method and formula to graphing in three dimensions, considering the z-axis and supplementing with an example problem.
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