Part A: Use complete sentences to explain the special relationship between the trigonometric ratios of triangles XYZ and ACB. You must show all work and calculations to receive full credit. (5 points) Part B: Explain how to find the measures of segments CB and AB. You must show all work and calculations to receive full credit. (5 points)

The first part of the question is missing. Here it is;

Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and cos X = 2.5/5.59

A)ΔXYZ and ΔACB are similar triangles because corresponding sides of both triangles are proportional while corresponding angles of both triangles are congruent.

B)AB = 11.18

CB = 10

Step-by-step explanation:

A) We are told that ΔXYZ was dilated by a scale factor of 2. This means that it was stretched by a factor of 2.

From the image of both triangles, we can see that;

∠Y = ∠C = 90º

Since ΔXYZ is a right angle triangle and it was only dilated by a factor of 2 to get ΔACB which is also a right angle triangles, then we can infer that;

∠X = ∠A

Now, in ΔXYZ, since ∠Y = 90°, then;

∠X + ∠Z = 90° (since sum of angles in a triangle is 180°)

Thus, ∠Z = 90° - ∠X

This means that ∠X and ∠Z are complementary angles.

Applying the same to ΔABC, we can say that;

∠B = 90° - ∠A

Thus, we can say that ∠A and ∠B are complementary angles.

Since they are similar triangles, we can say that; ∠Z = ∠B

From both triangles;

Since ∠X = ∠A ; ∠Z = ∠B and ∠Y = ∠C = 90º, it means we can conclude that ΔXYZ and ΔACB are similar triangles because corresponding sides of both triangles are proportional while corresponding angles of both triangles are congruent.

B) We are given;

cos ∠X = 2.5/5.59

In ΔXYZ;

cos ∠X = XY/XZ

Thus;

XY = 2.5 (the adjacent side)

XZ = 5.59 (hypotenuse)

From pythagoras theorem, we can find YZ

YZ = √(5.59² - 2.5²)

YZ = √24.9981

YZ ≈ 5

Now, since ΔXYZ was dilated by a factor of 2,it means that by corresponding sides;

CB/YZ = 2

Thus;

CB = 2YZ

CB = 2 × 5 =

CB = 10

Likewise;

AB/XZ = 2

AB = 2XZ

AB = 2 × 5.59

AB = 11.18

when functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, usually being just sets of points. these won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. small sets of points are generally the simplest sorts of relations, so your book starts with those.

step-by-step explanation:

state the domain and range of the following relation. is the relation a function?

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

the above list of points, being a relationship between certain x's and certain y's, is a relation. the domain is all the x-values, and the range is all the y-values. to give the domain and the range, i just list the values without duplication:

domain: {2, 3, 4, 6}

range: {–3, –1, 3, 6}

(it is customary to list these values in numerical order, but it is not required. sets are called "unordered lists", so you can list the numbers in any order you feel like. just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.)

while the given set does indeed represent a relation (because x's and y's are being related to each other), the set they gave me contains two points with the same x-value: (2, –3) and (2, 3). since x = 2 gives me two possible destinations (that is, two possible y-values), then this relation is not a function.

note that all i had to do to check whether the relation was a function was to look for duplicate x-values. if you find any duplicate x-values, then the different y-values mean that you do not have a function. remember: for a relation to be a function, each x-value has to go to one, and only one, y-value.

state the domain and range of the following relation. is the relation a function?

{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

i'll just list the x-values for the domain and the y-values for the range: