 6.1: What do you think is the most important or useful circle property y...
 6.2: How can you find the center of a circle with a compass and a straig...
 6.3: What does the path of a satellite have to do with the Tangent Conje...
 6.4: Explain the difference between the degree measure of an arc and its...
 6.5: Solve Exercises 519. If the exercise uses the = sign, answer in ter...
 6.6: Solve Exercises 519. If the exercise uses the = sign, answer in ter...
 6.7: Solve Exercises 519. If the exercise uses the = sign, answer in ter...
 6.8: Solve Exercises 519. If the exercise uses the = sign, answer in ter...
 6.9: Solve Exercises 519. If the exercise uses the = sign, answer in ter...
 6.10: Solve Exercises 519. If the exercise uses the = sign, answer in ter...
 6.11: Solve Exercises 519. If the exercise uses the = sign, answer in ter...
 6.12: Solve Exercises 519. If the exercise uses the = sign, answer in ter...
 6.13: Solve Exercises 519. If the exercise uses the = sign, answer in ter...
 6.14: Solve Exercises 519. If the exercise uses the = sign, answer in ter...
 6.15: Solve Exercises 519. If the exercise uses the = sign, answer in ter...
 6.16: Solve Exercises 519. If the exercise uses the = sign, answer in ter...
 6.17: Solve Exercises 519. If the exercise uses the = sign, answer in ter...
 6.18: Solve Exercises 519. If the exercise uses the = sign, answer in ter...
 6.19: Solve Exercises 519. If the exercise uses the = sign, answer in ter...
 6.20: On her latest archaeological dig, Ertha Diggs has unearthed a porti...
 6.21: Construction Construct a scalene obtuse triangle. Construct the cir...
 6.22: Construction Construct a scalene acute triangle. Construct the insc...
 6.23: Construction Construct a rectangle. Is it possible to construct the...
 6.24: Construction Construct a rhombus. Is it possible to construct the c...
 6.25: Find the equation of the line tangent to circle S centered at (1, 1...
 6.26: Find the center of the circle passing through the points (7, 5), (0...
 6.27: Rashid is an apprentice on a road crew for a civil engineer. He nee...
 6.28: Melanie rides the merrygoround on her favorite horse on the outer...
 6.29: Read the Geography Connection below. Given that the polar radius of...
 6.30: While talking to his friend Tara on the phone, Dmitri sees a lightn...
 6.31: King Arthur wishes to seat all his knights at a round table. He ins...
 6.32: If the circular moat should have been a circle of radius 10 meters ...
 6.33: The part of a circle enclosed by a central angle and the arc it int...
 6.34: In Exercises 3456, identify the statement as true or false. For eac...
 6.35: In Exercises 3456, identify the statement as true or false. For eac...
 6.36: In Exercises 3456, identify the statement as true or false. For eac...
 6.37: In Exercises 3456, identify the statement as true or false. For eac...
 6.38: In Exercises 3456, identify the statement as true or false. For eac...
 6.39: In Exercises 3456, identify the statement as true or false. For eac...
 6.40: In Exercises 3456, identify the statement as true or false. For eac...
 6.41: In Exercises 3456, identify the statement as true or false. For eac...
 6.42: In Exercises 3456, identify the statement as true or false. For eac...
 6.43: In Exercises 3456, identify the statement as true or false. For eac...
 6.44: In Exercises 3456, identify the statement as true or false. For eac...
 6.45: In Exercises 3456, identify the statement as true or false. For eac...
 6.46: In Exercises 3456, identify the statement as true or false. For eac...
 6.47: In Exercises 3456, identify the statement as true or false. For eac...
 6.48: In Exercises 3456, identify the statement as true or false. For eac...
 6.49: In Exercises 3456, identify the statement as true or false. For eac...
 6.50: In Exercises 3456, identify the statement as true or false. For eac...
 6.51: In Exercises 3456, identify the statement as true or false. For eac...
 6.52: In Exercises 3456, identify the statement as true or false. For eac...
 6.53: In Exercises 3456, identify the statement as true or false. For eac...
 6.54: In Exercises 3456, identify the statement as true or false. For eac...
 6.55: In Exercises 3456, identify the statement as true or false. For eac...
 6.56: In Exercises 3456, identify the statement as true or false. For eac...
 6.57: Find the measure of each lettered angle in the diagram below.
 6.58: Developing Proof In Exercises 5860, from the information given, det...
 6.59: Developing Proof In Exercises 5860, from the information given, det...
 6.60: Developing Proof In Exercises 5860, from the information given, det...
 6.61: Adventurer Dakota Davis has uncovered a piece of triangular tile fr...
 6.62: Circle O has a radius of 24 inches. Find the measure and the length...
 6.63: Developing Proof and EC are tangent to the circle, and AB = CD. Fin...
 6.64: Use your protractor to draw and label a pair of supplementary angle...
 6.65: Find the function rule f (n) of this sequence and find the 20th term.
 6.66: The design at right shows three hares joined by three ears, althoug...
 6.67: Construction Construct a rectangle whose length is twice its width.
 6.68: If AB = 15 cm, C is the midpoint of AB , D is the midpoint of AC, a...
 6.69: Draw the next shape in this pattern.
 6.70: Construction Construct any triangle. Then construct its centroid.
Solutions for Chapter 6: Discovering and Proving Circle Properties
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 6: Discovering and Proving Circle Properties
Get Full SolutionsSince 70 problems in chapter 6: Discovering and Proving Circle Properties have been answered, more than 23381 students have viewed full stepbystep solutions from this chapter. Chapter 6: Discovering and Proving Circle Properties includes 70 full stepbystep solutions. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4. Discovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. This expansive textbook survival guide covers the following chapters and their solutions.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Iterative method.
A sequence of steps intended to approach the desired solution.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.