 2.1.1: On the real number line the origin is assigned the number .
 2.1.2: If and 5 are the coordinates of two points on the real number line,...
 2.1.3: If 3 and 4 are the legs of a right triangle, the hypotenuse is
 2.1.4: Use the converse of the Pythagorean Theorem to show that a triangle...
 2.1.5: The area A of a triangle whose base is b and whose altitude is h is A
 2.1.6: True or False Two triangles are congruent if two angles and the inc...
 2.1.7: If are the coordinates of a point P in the xyplane, then x is call...
 2.1.8: The coordinate axes divide the xyplane into four sections called
 2.1.9: If three distinct points P, Q, and R all lie on a line and if then ...
 2.1.10: True or False The distance between two points is sometimes a negati...
 2.1.11: True or False The point lies in quadrant IV of the Cartesian plane.
 2.1.12: True or False The midpoint of a line segment is found by averaging ...
 2.1.13: In 13 and 14, plot each point in the xyplane. Tell in which quadra...
 2.1.14: In 13 and 14, plot each point in the xyplane. Tell in which quadra...
 2.1.15: Plot the points and . Describe the set of all points of the form wh...
 2.1.16: Plot the points , and . Describe the set of all points of the form ...
 2.1.17: In 1728, find the distance between the points and P2 P . d1P1 1 , P22
 2.1.18: In 1728, find the distance between the points and P2 P . d1P1 1 , P22
 2.1.19: In 1728, find the distance between the points and P2 P . d1P1 1 , P22
 2.1.20: In 1728, find the distance between the points and P2 P . d1P1 1 , P22
 2.1.21: In 1728, find the distance between the points and P2 P . d1P1 1 , P...
 2.1.22: In 1728, find the distance between the points and P2 P . d1P1 1 , P...
 2.1.23: In 1728, find the distance between the points and P2 P . d1P1 1 , P...
 2.1.24: In 1728, find the distance between the points and P2 P . d1P1 1 , P...
 2.1.25: In 1728, find the distance between the points and P2 P . d1P1 1 , P...
 2.1.26: In 1728, find the distance between the points and P2 P . d1P1 1 , P...
 2.1.27: In 1728, find the distance between the points and P2 P . d1P1 1 , P...
 2.1.28: In 1728, find the distance between the points and P2 P . d1P1 1 , P...
 2.1.29: In 2934, plot each point and form the triangle ABC. Verify that the...
 2.1.30: In 2934, plot each point and form the triangle ABC. Verify that the...
 2.1.31: In 2934, plot each point and form the triangle ABC. Verify that the...
 2.1.32: In 2934, plot each point and form the triangle ABC. Verify that the...
 2.1.33: In 2934, plot each point and form the triangle ABC. Verify that the...
 2.1.34: In 2934, plot each point and form the triangle ABC. Verify that the...
 2.1.35: In 3542, find the midpoint of the line segment joining the points a...
 2.1.36: In 3542, find the midpoint of the line segment joining the points a...
 2.1.37: In 3542, find the midpoint of the line segment joining the points a...
 2.1.38: In 3542, find the midpoint of the line segment joining the points a...
 2.1.39: In 3542, find the midpoint of the line segment joining the points a...
 2.1.40: In 3542, find the midpoint of the line segment joining the points a...
 2.1.41: In 3542, find the midpoint of the line segment joining the points a...
 2.1.42: In 3542, find the midpoint of the line segment joining the points a...
 2.1.43: If the point is shifted 3 units to the right and 2 units down, what...
 2.1.44: If the point is shifted 2 units to the left and 4 units up, what ar...
 2.1.45: Find all points having an xcoordinate of 3 whose distance from the...
 2.1.46: Find all points having a ycoordinate of whose distance from the po...
 2.1.47: Find all points on the xaxis that are 6 units from the point.
 2.1.48: Find all points on the yaxis that are 6 units from the point.
 2.1.49: The midpoint of the line segment from P1 to P2 is . If, what is ?
 2.1.50: The midpoint of the line segment from to is . Ifwhat is ?
 2.1.51: Geometry The medians of a triangle are the line segments from each ...
 2.1.52: Geometry An equilateral triangle is one in which all three sides ar...
 2.1.53: Geometry Find the midpoint of each diagonal of a square with side o...
 2.1.54: Geometry Verify that the points (0, 0), (a, 0), and are the vertice...
 2.1.55: In 5558, find the length of each side of the triangle determined by...
 2.1.56: In 5558, find the length of each side of the triangle determined by...
 2.1.57: In 5558, find the length of each side of the triangle determined by...
 2.1.58: In 5558, find the length of each side of the triangle determined by...
 2.1.59: Baseball A major league baseball diamond is actually a square, 90 f...
 2.1.60: Little League Baseball The layout of a Little League playing field ...
 2.1.61: Baseball Refer to 59. Overlay a rectangular coordinate system on a ...
 2.1.62: Little League Baseball Refer to 60. Overlay a rectangular coordinat...
 2.1.63: Distance between Moving Objects A Dodge Neon and a Mack truck leave...
 2.1.64: Distance of a Moving Object from a Fixed Point A hotair balloon, h...
 2.1.65: Drafting Error When a draftsman draws three lines that are to inter...
 2.1.66: Net Sales The figure illustrates how net sales of WalMart Stores, ...
 2.1.67: Poverty Threshold Poverty thresholds are determined by the U.S. Cen...
 2.1.68: Write a paragraph that describes a Cartesian plane. Then write a se...
Solutions for Chapter 2.1: The Distance and Midpoint Formulas
Full solutions for College Algebra  9th Edition
ISBN: 9780321716811
Solutions for Chapter 2.1: The Distance and Midpoint Formulas
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra, edition: 9. Chapter 2.1: The Distance and Midpoint Formulas includes 68 full stepbystep solutions. Since 68 problems in chapter 2.1: The Distance and Midpoint Formulas have been answered, more than 32863 students have viewed full stepbystep solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9780321716811.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Outer product uv T
= column times row = rank one matrix.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Solvable system Ax = b.
The right side b is in the column space of A.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.