 12.2.1: For Exercises 19, find each length or angle measure accurate to the...
 12.2.2: For Exercises 19, find each length or angle measure accurate to the...
 12.2.3: For Exercises 19, find each length or angle measure accurate to the...
 12.2.4: For Exercises 19, find each length or angle measure accurate to the...
 12.2.5: For Exercises 19, find each length or angle measure accurate to the...
 12.2.6: For Exercises 19, find each length or angle measure accurate to the...
 12.2.7: For Exercises 19, find each length or angle measure accurate to the...
 12.2.8: For Exercises 19, find each length or angle measure accurate to the...
 12.2.9: For Exercises 19, find each length or angle measure accurate to the...
 12.2.10: According to a Chinese legend from the Han dynasty (206 B.C.E.220 C...
 12.2.11: Benny is flying a kite directly over his friend Frank, who is 125 m...
 12.2.12: Application The angle of elevation from a ship to the top of a 42m...
 12.2.13: Application A salvage ships sonar locates wreckage at a 12 angle of...
 12.2.14: Application A ships officer sees a lighthouse at a 42 angle to the ...
 12.2.15: Application A meteorologist shines a spotlight vertically onto the ...
 12.2.16: Application Meteorologist Wendy Stevens uses a theodolite (an angle...
 12.2.17: For Exercises 1719, find the measure of each angle to the nearest d...
 12.2.18: For Exercises 1719, find the measure of each angle to the nearest d...
 12.2.19: For Exercises 1719, find the measure of each angle to the nearest d...
 12.2.20: Solve for x. 21. a. 4.7 b. 8 c. 0.3736 d. 0.9455
 12.2.21: Find x and y.
 12.2.22: A 3by5by6 cm block of wood is dropped into a cylindrical contai...
 12.2.23: Scalene triangle ABC has altitudes AX , BY, and CZ . If AB > BC > A...
 12.2.24: In the diagram at right, PT and PS are tangent to circle O at point...
 12.2.25: Points S and Q, shown at right, are consecutive vertices of square ...
Solutions for Chapter 12.2: Problem Solving with Right Triangles
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 12.2: Problem Solving with Right Triangles
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Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

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

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

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.