 9.1.1: In Exercises 111, find each missing length. All measurements are in...
 9.1.2: In Exercises 111, find each missing length. All measurements are in...
 9.1.3: In Exercises 111, find each missing length. All measurements are in...
 9.1.4: In Exercises 111, find each missing length. All measurements are in...
 9.1.5: In Exercises 111, find each missing length. All measurements are in...
 9.1.6: In Exercises 111, find each missing length. All measurements are in...
 9.1.7: In Exercises 111, find each missing length. All measurements are in...
 9.1.8: In Exercises 111, find each missing length. All measurements are in...
 9.1.9: In Exercises 111, find each missing length. All measurements are in...
 9.1.10: In Exercises 111, find each missing length. All measurements are in...
 9.1.11: In Exercises 111, find each missing length. All measurements are in...
 9.1.12: A baseball infield is a square, each side measuring 90 feet. To the...
 9.1.13: The diagonal of a square measures 32 meters. What is the area of th...
 9.1.14: What is the length of the diagonal of a square whose area is 64 cm2 ?
 9.1.15: The lengths of the three sides of a right triangle are consecutive ...
 9.1.16: A rectangular garden 6 meters wide has a diagonal measuring 10 mete...
 9.1.17: Developing Proof One very famous proof of the Pythagorean Theorem i...
 9.1.18: Developing Proof Is ABC XYZ ? Explain your reasoning
 9.1.19: The two quadrilaterals whose areas are given are squares. Find the ...
 9.1.20: Give the vertex arrangement for the 2uniform tessellation.
 9.1.21: Developing Proof Explain why m + n = 120.
 9.1.22: Developing Proof Calculate each lettered angle, measure, or arc. EF...
 9.1.23: Two tugboats are pulling a container ship into the harbor. They are...
Solutions for Chapter 9.1: The Theorem of Pythagoras
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 9.1: The Theorem of Pythagoras
Get Full SolutionsDiscovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. Chapter 9.1: The Theorem of Pythagoras includes 23 full stepbystep solutions. Since 23 problems in chapter 9.1: The Theorem of Pythagoras have been answered, more than 23233 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.