 1.5.1: For Exercises 14, match the term on the left with its figure on the...
 1.5.2: For Exercises 14, match the term on the left with its figure on the...
 1.5.3: For Exercises 14, match the term on the left with its figure on the...
 1.5.4: For Exercises 14, match the term on the left with its figure on the...
 1.5.5: For Exercises 59, sketch and label the figure. Mark the figures. Is...
 1.5.6: For Exercises 59, sketch and label the figure. Mark the figures. Sc...
 1.5.7: For Exercises 59, sketch and label the figure. Mark the figures. Is...
 1.5.8: For Exercises 59, sketch and label the figure. Mark the figures. Tw...
 1.5.9: For Exercises 59, sketch and label the figure. Mark the figures. Tw...
 1.5.10: Use your ruler and protractor to draw an isosceles acute triangle w...
 1.5.11: Use your ruler and protractor to draw an isosceles obtuse triangle ...
 1.5.12: For Exercises 1214, use the graphs below. Can you find more than on...
 1.5.13: For Exercises 1214, use the graphs below. Can you find more than on...
 1.5.14: For Exercises 1214, use the graphs below. Can you find more than on...
 1.5.15: Use your ruler and protractor to draw a triangle with one side 9 cm...
 1.5.16: Use your ruler and protractor to draw a triangle with one angle mea...
 1.5.17: For Exercises 1721, tell whether the statement is true or false. Fo...
 1.5.18: For Exercises 1721, tell whether the statement is true or false. Fo...
 1.5.19: For Exercises 1721, tell whether the statement is true or false. Fo...
 1.5.20: For Exercises 1721, tell whether the statement is true or false. Fo...
 1.5.21: For Exercises 1721, tell whether the statement is true or false. Fo...
 1.5.22: Use the ordered pair rule (x, y) (x + 1, y 3) to relocate the four ...
 1.5.23: Suppose a set of thin rods is glued together into a triangle as sho...
 1.5.24: For Exercises 2426, sketch and carefully label the figure. Mark the...
 1.5.25: For Exercises 2426, sketch and carefully label the figure. Mark the...
 1.5.26: For Exercises 2426, sketch and carefully label the figure. Mark the...
Solutions for Chapter 1.5: Triangles
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 1.5: Triangles
Get Full SolutionsDiscovering 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. Since 26 problems in chapter 1.5: Triangles have been answered, more than 25054 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. Chapter 1.5: Triangles includes 26 full stepbystep solutions.

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

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.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

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).

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.