Podcast
Questions and Answers
A ___ is a geometric solid formed by a circular base and a curved surface that connects the base to a vertex.
A ___ is a geometric solid formed by a circular base and a curved surface that connects the base to a vertex.
cone
Two cones placed vertex to vertex is called a ___.
Two cones placed vertex to vertex is called a ___.
double cone
A ___ is one of two pieces of a double cone divided at the vertex.
A ___ is one of two pieces of a double cone divided at the vertex.
nappe
A ___ is the intersection of a plane with one or both nappes of a double cone.
A ___ is the intersection of a plane with one or both nappes of a double cone.
Label each conic section by writing its name on the blank.
Label each conic section by writing its name on the blank.
Determine the x-value of the solution of the linear system by using Cramer's Rule; then choose the correct answer. [1 3 | 1] [4 8 | 0]
Determine the x-value of the solution of the linear system by using Cramer's Rule; then choose the correct answer. [1 3 | 1] [4 8 | 0]
Determine the x-value of the solution of the linear system by using Cramer's Rule; then choose the correct answer. [1 2 | -3] [2 -3 | 8]
Determine the x-value of the solution of the linear system by using Cramer's Rule; then choose the correct answer. [1 2 | -3] [2 -3 | 8]
Determine the x-value of the solution of the linear system by using Cramer's Rule; then choose the correct answer. [2 3 | 1] [-3 -4 | -3]
Determine the x-value of the solution of the linear system by using Cramer's Rule; then choose the correct answer. [2 3 | 1] [-3 -4 | -3]
Match the sentences with the correct words.
Match the sentences with the correct words.
Match the descriptions with the formulas.
Match the descriptions with the formulas.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Find the midpoint of AB¯.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Find the midpoint of AB¯.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Find the midpoint of BC¯.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Find the midpoint of BC¯.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Find slope of mAB¯.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Find slope of mAB¯.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Find slope of mBC¯.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Find slope of mBC¯.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯, mAB=-2, mBC=2. Find slope of PAB¯, perpendicular to AB¯.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯, mAB=-2, mBC=2. Find slope of PAB¯, perpendicular to AB¯.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯, mAB=-2, mBC=2. Find slope of PBC¯, perpendicular to BC¯.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯, mAB=-2, mBC=2. Find slope of PBC¯, perpendicular to BC¯.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯, mPAB=12, mPBC=-12. Find standard form equation of line perpendicular to AB¯ through the midpoint.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯, mPAB=12, mPBC=-12. Find standard form equation of line perpendicular to AB¯ through the midpoint.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯, mPAB=12, mPBC=-12. Find standard form equation of line perpendicular to BC¯ through the midpoint.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯, mPAB=12, mPBC=-12. Find standard form equation of line perpendicular to BC¯ through the midpoint.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Use Gaussian elimination to find the intersection of the lines, the center of the circle containing points.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Use Gaussian elimination to find the intersection of the lines, the center of the circle containing points.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Use the center point and point A to find the radius of the circle.
Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Use the center point and point A to find the radius of the circle.
Given: A(2,1), B(0,5), C(-1,2), AB¯ and BC¯, lines x-2y=10 and -x-2y=6 through the midpoints of AB¯ and BC¯ and perpendicular to them. Use Gaussian elimination to find the intersection of the lines.
Given: A(2,1), B(0,5), C(-1,2), AB¯ and BC¯, lines x-2y=10 and -x-2y=6 through the midpoints of AB¯ and BC¯ and perpendicular to them. Use Gaussian elimination to find the intersection of the lines.
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Study Notes
Conic Sections
- Cone: A geometric solid with a circular base and a curved surface connecting the base to a vertex.
- Double Cone: Formed by two cones placed vertex to vertex.
- Nappe: Each of the two pieces of a double cone, divided at the vertex.
- Conic Section: The intersection of a plane with one or both nappes of a double cone.
- Types of Conic Sections: Parabola, circle, ellipse, hyperbola.
Linear Systems and Cramer's Rule
- Cramer's Rule is used to determine the solutions of linear systems.
- Example solutions include x-values of -2, 1, and 5 for specified systems.
Geometric Concepts
- Locus of Points: A set of points whose locations satisfy a specific description.
- Circle Definition: The locus of points equidistant from a center point.
- Axis of a Cone: A segment extending from the vertex to the center of the base.
Midpoints and Slopes
- Midpoints can be calculated for line segments AB and BC.
- Slope of the lines is determined from the coordinates of points and calculated for perpendicular lines.
Circle Equations
- The equation of a circle can be derived using its center and radius.
- Standard forms include ((x - h)^2 + (y - k)^2 = r^2).
Inverses of Functions
- The inverses of linear functions can be calculated, for example, (f(x) = 5x + 10) results in (y = \frac{1}{5}x - 2).
Ellipses Properties
- Ellipse Definition: A locus of points such that the sum of the distances to two foci is constant.
- Axes of Ellipses: Major axis (line through vertices) and minor axis (line through co-vertices).
- Vertices and Co-vertices: Determined based on the standard form of the ellipse's equation.
Standard Form Equations
- Circle: ((x-h)^2 + (y-k)^2 = r^2)
- Ellipse: Depends on orientation (horizontal or vertical).
Hyperbolas
- Definition: A locus of points where the difference of distances to two foci is constant.
- Axes: Transverse (joining the vertices) and conjugate (perpendicular through the center).
- Asymptotes: Lines that the hyperbola approaches but never touches.
Matrices
- Operations with matrices include addition, multiplication, scalar multiplication, and finding inverses. The structure of matrices changes with these operations.
Example Values
- For the ellipse given by (x^2/49 + y^2/25 = 1):
- a = 7, b = 5, center (0, 0), vertices (-7, 0), (7, 0), co-vertices (0, -5), (0, 5).
Important Formulas
- Distance Formula: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2})
- Midpoint Formula: ((x_1+x_2 ,/, 2, y_1+y_2 ,/, 2))
Circle and Ellipses Analysis
- The center and radius of circles or ellipses can be extracted from given equations through completing the square or standard form transformation.
Transformations
- Hyperbola equations can be transformed into standard form through algebraic manipulation.
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