Algebra II Unit 4 Flashcards

Questions and Answers

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 ___.

double cone

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.

conic section Signup and view all the answers

Label each conic section by writing its name on the blank.

Ellipse Signup and view all the answers

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]

-2 Signup and view all the answers

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]

1 Signup and view all the answers

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]

5 Signup and view all the answers

Match the sentences with the correct words.

A(n) __________ is a set of points whose location satisfies a particular description. = locus of points A(n) __________ is the locus of points in a plane that are equidistant from one point, called the center. = circle Points that do not lie on the same straight line with other points are __________. = noncollinear points The __________ is a segment that extends from the vertex of a cone to the center of the base. = axis of a cone Signup and view all the answers

Match the descriptions with the formulas.

(A.) m = (y₂-y₁)/(x₂-x₁) = slope formula (D.) (x-x₁)² + (y-y₁)² = r² = circle formula (B.) (x₁+x₂)/2, (y₁+y₂)/2 = midpoint formula (E.) m = (y-y₁)/(x-x₁) = point-slope formula (C.) d = √((x₂-x₁)² + (y₂-y₁)²) = distance formula Signup and view all the answers

Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Find the midpoint of AB¯.

(6, -2) Signup and view all the answers

Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Find the midpoint of BC¯.

(6, -6) Signup and view all the answers

Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Find slope of mAB¯.

-2 Signup and view all the answers

Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯. Find slope of mBC¯.

2 Signup and view all the answers

Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯, mAB=-2, mBC=2. Find slope of PAB¯, perpendicular to AB¯.

½ Signup and view all the answers

Given: A(5,0), B(7,-4), C(5,-8), AB¯ and BC¯, mAB=-2, mBC=2. Find slope of PBC¯, perpendicular to BC¯.

-½ Signup and view all the answers

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.

x-2y=10 Signup and view all the answers

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.

-x-2y=6 Signup and view all the answers

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.

(2, -4) Signup and view all the answers

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.

5 Signup and view all the answers

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.

(1, 3) Signup and view all the answers

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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Description

Test your understanding of geometric solids with this flashcard quiz from Algebra II Unit 4. Each card presents a definition or term related to cones and double cones, providing a quick and effective way to reinforce your knowledge. Perfect for students looking to ace their geometry concepts!