Find the constant of proportionality k if y = 27 when x = 3.
k =

Using the k from above find x given that y = 9.
y =

Answer:

a = 3√2;

b = 3

Step-by-step explanation:

Use trigonometry:

[tex] \tan(45°) = \frac{b}{3} [/tex]

Cross-multiply to find b:

[tex]b = 3 \times \tan(45°) = 3 \times 1 = 3[/tex]

Use the Pythagorean theorem to find a:

[tex] {a}^{2} = {3}^{2} + {b}^{2} [/tex]

[tex] {a}^{2} = {3}^{2} + {3}^{2} = 9 + 9 = 18[/tex]

[tex]a > 0[/tex]

[tex]a = \sqrt{18} = \sqrt{9 \times 2} = 3 \sqrt{2} [/tex]

The identity 1/csc x+1 - 1/csc x-1 = [tex]-2tan^2 x[/tex] is verified and correct.

To verify the identity:

[tex]\\\frac{1}{csc x+1} - \frac{1}{csc x-1} = -2tan^2 x[/tex]

Starting with the reciprocal identity, we can say:

csc x = [tex]\frac{1}{sin x}[/tex]

So we have:

[tex]1/(1/sin x + 1) - 1/(1/sin x - 1) = -2tan^2 x[/tex]

We need to identify a common denominator in order to simplify the left side of the equation. The common denominator is:

[tex](1/sin x + 1)(1/sin x - 1) = (1 - sin x)/(sin x)^2[/tex]

As a result, we can change the left side of the equation to read:

[tex][(1 - sin x)/(sin x)^2] [(sin x - 1)/(sin x + 1)] - [(1 - sin x)/(sin x)^2] [(sin x + 1)/(sin x - 1)][/tex]

Simplifying this expression by multiplying the numerators and denominators, we get:

[tex](1 - sin x)(sin x - 1) - (1 - sin x)(sin x + 1) / (sin x + 1)(sin x - 1)(sin x)^2[/tex]

Expanding the brackets and simplifying, we get:

[tex]-(2sin^2 x - 2sin x) / (sin x + 1)(sin x - 1)(sin x)^2[/tex]

Factor out -2sin x from the numerator:

[tex]-2sin x(sin x - 1) / (sin x + 1)(sin x - 1)(sin x)^2[/tex]

Simplifying, we get:

[tex]-2sin x / (sin x + 1)(sin x)^2[/tex]

Now, we can use the identity:

[tex]tan^2 x = sec^2 x - 1 = (1/cos^2 x) - 1 = sin^2 x / (1 - sin^2 x)[/tex]

Simplifying, we get:

[tex]sin^2 x = tan^2 x (1 - tan^2 x)[/tex]

When we add this to the initial equation, we obtain:

[tex]-2sin x / (sin x + 1)(sin x)^2 = -2tan^2 x(sin x)/(sin x + 1)[/tex]

Now, we can use the identity:

sin x / (sin x + 1) = 1 - 1/(sin x + 1)

Simplifying, we get:

[tex]-2tan^2 x(sin x)/(sin x + 1) = -2tan^2 x + 2tan^2 x / (sin x + 1)[/tex]

When we add this to the initial equation, we obtain:[tex]-2tan^2 x + 2tan^2 x / (sin x + 1) = -2tan^2 x[/tex]

Simplifying, we get:

[tex]-2tan^2 x = -2tan^2 x[/tex]

Therefore, the identity is verified.

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Since the two equations represent the same line, there are infinitely many solutions to the system.

An equation is a mathematical statement that asserts the equality of two expressions. It consists of two sides, each containing one or more terms, separated by an equal sign. The terms may contain variables, constants, and mathematical operators such as addition, subtraction, multiplication, division, exponentiation, and roots. Equations are used to represent relationships between variables and to solve problems in mathematics, science, engineering, and many other fields. They can be classified based on their degree, number of variables, and the types of functions they involve. Linear equations, for example, involve only first-degree terms and are used to represent straight lines; quadratic equations involve second-degree terms and are used to represent parabolas.

Here,

To determine whether the graphs of the equations are identical lines, parallel lines, or lines intersecting at a single point, we can first put the equations in slope-intercept form (y = mx + b), where m is the slope of the line and b is the y-intercept.

10y + 8x = 10

10y = -8x + 10

y = (-4/5)x + 1

5y - 5 = -4x

5y = -4x + 5

y = (-4/5)x + 1

Notice that the two equations have the same slope of -4/5 and the same y-intercept of 1. Therefore, the two lines are identical and intersect at every point on the line.

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A) Let X be the amount of time it takes Capt. Rigg to land a plane from announcement to touch down. We know that X has a uniform distribution from 0 to 25 minutes. The probability that it takes at least 5 minutes to land the plane is equal to the probability that X is greater than or equal to 5:

P(X ≥ 5) = (25-5)/(25-0) = 20/25 = 0.8

So the probability that, for a randomly selected flight, it takes Capt. Rigg at least 5 minutes to land the plane after the announcement is 0.8.

b) Since X has a continuous uniform distribution, the probability that it takes exactly 15 minutes to land a plane is 0.

The net profit can be calculated by subtracting the production costs from the total revenue generated by selling tickets. Since each ticket was sold for a fixed price, we can assume that the relationship between the net profit and the number of tickets sold is linear.

We know that when 200 tickets were sold, the net profit was $12,000, which means that the slope of the linear function is:

[tex]\text{slope = (net profit at 200 tickets - net profit at 0 tickets)} \div (200 - 0)[/tex]

[tex]\text{slope} = (\$12,000 - \$1,200) \div 200[/tex]

[tex]\text{slope} = \$55[/tex]

The y-intercept of the linear function represents the net profit when no tickets have been sold, which is equal to the negative of the production costs:

[tex]\text{y-intercept} = -\$1,200[/tex]

Therefore, the equation of the linear function is:

[tex]\text{y} = \$55x - \$1,200[/tex]

where x is the number of tickets sold and y is the net profit in dollars.

The graph of this function is an increasing linear function in quadrant 1 with a positive y-intercept, which is choice A. Therefore, the answer is choice A: graph of an increasing linear function in quadrant 1 with a positive y-intercept.

Answer: 20

Step-by-step explanation:

The answer of the given question based on the relation is R = {(1,1), (3,3), (5,5), (2,2), (4,4), (6,6), (7,7), (1,3), (3,1), (1,5), (5,1), (1,7), (7,1), (2,4), (4,2), (2,6), (6,2), (3,5), (5,3), (3,7), (7,3), (4,6), (6,4), (5,7), (7,5)}.

Set listing notation is a way to represent a set by listing all of its elements between curly braces { }. If a set has a large or infinite number of elements, it may not be practical to list all the elements. In such cases, we can use set-builder notation to describe the set using a condition or a rule.

The "has the same parity as" relation on set A means that any two elements in A are related if they have the same parity (either both even or both odd).

So, we can write the relation R as a set of ordered pairs where each pair consists of two elements that have the same parity.

R = {(1,1), (3,3), (5,5), (2,2), (4,4), (6,6), (7,7), (1,3), (3,1), (1,5), (5,1), (1,7), (7,1), (2,4), (4,2), (2,6), (6,2), (3,5), (5,3), (3,7), (7,3), (4,6), (6,4), (5,7), (7,5)}

In this set, the ordered pair (x, y) indicates that x and y have the same parity. For example, (1, 3) is in R because 1 and 3 are both odd. Similarly, (2, 4) is in R because 2 and 4 are both even.

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Answer:

that's the slope of the y intercept

The equation that models the total amount Anna saves, y, in terms of the number of days she adds to her savings, x, can be represented as y = 2x + 125, where 2x represents the total amount she saves from her allowance and 125 represents the initial amount she saved.

This is a linear equation in slope-intercept form, where the slope of the line is 2, indicating that she saves $2 per day, and the y-intercept is 125, indicating that she had already saved $125 before starting to save from her allowance. To graph this equation, we can plot the y-intercept at (0, 125) and use the slope to find other points on the line. For example, after 5 days, she would have saved y = 2(5) + 125 = 135 dollars, giving us the point (5, 135). Plotting this point and connecting it with the y-intercept will give us a straight-line graph.

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According to the quadrilaterals, the proof perpendicularity and congruence are stated below.

Proof #7:

Statement | Reasons

XY | ZW | Given

XW bisects ZY | Given

ZR ≅ RY | Definition of segment bisector

∠XRY ≅ ∠RW | Alternate interior angles theorem

ΔXRY ≅ ΔWRZ | AAS ≅ theorem

∠XYR ≅ ∠WZR | Definition of segment bisector and corresponding parts of congruent triangles

Proof #8:

Statements | Reasons

EF ≅ HL | Given

∠PER ≅ ∠PHE | Given

∠EPF and ∠HPL are right angles | Definition of perpendicular lines

EP ≅ PH | Definition of perpendicular bisector

AEFP ≅ AHLP | SAS ≅ theorem

ΔEFP ≅ ΔHLP | Base angles converse theorem

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Image transcribed:

A Proof #7

Given: XY || ZW

XW bisects ZY

Prove: ΔXRY ≅ ΔWRZ

Statements | Reasons

1.                | 1.

2.                | 2.

3.                | 3.

4.                | 4.

5.                | 5.

6.                | 6.

XY || ZW,   Alternate Int. ∠ Theorem,   AAS ≅ Theorem

∠XRY ≅ ∠RW,    Def. of Segment Bisector,   ZR ≅ RY

ΔXRY ≅ ΔWRZ,    ∠XYR ≅ ∠WZR,    XW bisects ZY

ASA ≅ Theorem,   Given,     Vertical Angles ≅ Theorem

A Proof # 8

Given: EL ⊥ FH, ∠PEH ≅ ∠PHE

EF ≅ HL

Prove: ΔEFP ≅ ΔHLP

Statements | Reasons

1.                | 1.

2.                | 2.

3.                | 3.

4.                | 4.

5.                | 5.

6.                | 6.

EF ≅ HL,    AEFP ≅ AHLP,    EP ≅ PH,    ∠PER ≅ ∠PHE

HL ≅ Theorem,      SAS ≅ Theorem,      Base Angles Converse Theorem

Definition of ⊥ Lines,     ∠EPF and ∠HPL are right angles

EL ⊥ FH,        Given

The relation between time and depth is not prοpοrtiοnal as 2/100 ≠ 4/180 ≠ 6/260 ≠ 8/340.

Proportional relationships are relationships between two variables where their ratios are equivalent. Another way to think about them is that, in a proportional relationship, one variable is always a constant value times the other. That constant is known as the "constant of proportionality"

To solve questions that involve a table showing the relationship between two variables, such as depth and time in this case, you may want to consider the following steps:

A quick check οf table values shοws the relatiοnship is nοt prοpοrtiοnal:

100/2 = 50 ≠ 45 = 180/4

That is, the ratiοs οf table values are nοt cοnstant.

2/100 ≠ 4/180 ≠ 6/260 ≠ 8/340

Therefore, the relation between time and depth is not prοpοrtiοnal.

When the values are graphed, the line thrοugh the pοints dοes nοt intersect the οrigin. This is further indicatiοn the relatiοnship is nοt prοpοrtiοnal.

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Complete question:

Answer:

(D) 8 units to the left

Step-by-step explanation:

Took the test, got it right. FLVS

Value of side x in equilateral triangle is 6units.

An equilateral triangle is a type of triangle in which all three sides are of equal length. Equilateral triangles are also equiangular, meaning all three angles are of equal measure and are each 60 degrees. Because all three sides and angles are equal, equilateral triangles are symmetric about their center point, and their three medians, altitudes, and angle bisectors are all the same line.

Area of equilateral triangle = √3/4×a²

Given side length=12

So, area of equilateral triangle =√3/4×12²=62.35383

We know that

Area of triangle= ½×base×height

Given

Base=12

let height of triangle be=y

Area=½×12×y

62.35383=½×12×y

y=10.39

Using pythagoras' theorem,

c²=a²+b²

122=x²+10.392

x=√144-107.95

x=6

Hence, value of side x in equilateral triangle is 6units.

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The limit of (2n)! as n goes to 0 does not exist.

Limits are a fundamental concept in mathematics that describe the behavior of a function as the input variable approaches a certain value, either from one or both sides. They are used to determine values that a function can get arbitrarily close to but not necessarily equal to.

The limit of (2n)! as n goes to 0 does not exist, as the factorial function is not defined for non-negative integers less than 1. The factorial function is defined as the product of all positive integers up to and including the argument. Therefore, for non-negative integers less than 1, the factorial function is undefined. The limit does not exist because the function is undefined for the given value of n.

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Prove that Limit (2n)! as n goes to 0 does not exist.

Answer:

To find the slope of a line perpendicular to another line, we need to first find the slope of the given line. The equation of the given line is x + y = 3. We can rewrite this equation in slope-intercept form (y = mx + b) by solving for y: y = -x + 3 So the slope of the given line is -1. To find the slope of a line perpendicular to this line, we know that it will have a slope that is the negative reciprocal of -1, which is 1. Therefore, the slope of a line perpendicular to the line whose equation is x + y = 3 is 1.

Answer: 1

Step-by-step explanation:

The given equation x + y = 3 can be rearranged to slope-intercept form, which is y = -x + 3.

To find the slope of this line, we can see that the coefficient of x is -1. Therefore, the slope of the line is -1.

To find the slope of a line perpendicular to this line, we need to take the negative reciprocal of the slope of the given line.

The negative reciprocal of -1 is 1/1 or simply 1. Therefore, the slope of a line perpendicular to the line x + y = 3 is 1.

Answer:

D

Step-by-step explanation:

-6 is constant because it has a degree of 0

3x is linear because it has a degree of 1

[tex]4x^{2}[/tex] is quadratic because it has a degree of 2

So the answer is D

Answer: it makes 74 degree

Step-by-step explanation: This forms a right triangle. The 25-ft ladder is the hypotenuse of the right triangle. and the 7ft bottom of the ladder is the base of a triangle

For the angle where the ladder meets the ground, the

ground is the adjacent leg. The ladder is the hypotenuse.

Call the angle between the ladder and the ground angle A.

The trig ratio that relates the adjacent leg and the hypotenuse is cosine.

After answering the presented question, we can conclude that  area of Circle M, [tex]C = 2\pi [(17/\pi )^0.5] km\\[/tex]

A circle appears to be a two-dimensional component that is defined as the collection of all places in a jet that are equidistant from the hub. A circle is typically depicted with a capital "O" for the centre and a lower portion "r" for the radius, which represents the distance from the origin to any point on the circle. The formula 2r gives the girth (the distance from the centre of the circle), where (pi) is a proportionality constant about equal to 3.14159. The formula r2 computes the circumference of a circle, which relates to the amount of space inside the circle.

area of Circle M,

[tex]289 = \PI(x + 6)^2\\289/\PI = (x + 6)^2\\\sqrt(289/\pi ) = x + 6\\(17/\pi )^0.5 - 6 = x\\x = (17/\pi )^0.5 - 6 km\\C = \pi (2x + 12) km\\C = 2\pi (x + 6) km\\[/tex]

[tex]C = 2\pi [(17/\pi )^0.5] km\\[/tex]

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Answer: [tex]-6x^3+23x^2+59x-120[/tex]

Step-by-step explanation:

(2x – 3)(8 + 3x)( 5 - x)

To multiply this polynomial we can break it down into two steps.

First, lets multiply (2x – 3)(8 + 3x) from the polynomial (2x – 3)(8 + 3x)(5 - x).

Its best to rearrange the x in the parenthesis, so it can look and be much easier: (2x – 3)(8 + 3x)  ->  (2x – 3)(3x + 8)

Now lets multiply it out: (2x – 3)(3x + 8)

[tex]6x^2+16x-9x-24[/tex]

After multiplying it out, combine like terms:

[tex]6x^2+7x-24[/tex]

Now take this polynomial and multiply the remaining factor, (5 - x)

[tex](-x + 5)(6x^2+7x-24)\\\\-6x^3+-7x^2+24x+30x^2+35x-120[/tex]

Now lets combine like terms!

[tex]-6x^3+23x^2+59x-120[/tex]

So the answer is: [tex]-6x^3+23x^2+59x-120[/tex]

The function f(x) will be after shifting is g(x)= (x – 1)² + 1

The function f is used to move the graph of a function f(x) to the right by h units (x-h). This is because when we substitute x with x-h in f(x), we obtain f(x-h), meaning that we are substituting x with x+h in the initial function f. (x). Hence, if we wish to move the graph of f(x) = (x + 2)² + 1 three units to the right, we may do it by using the function f(x-3) to move the graph three units to the right.

g(x)=(x-3+2)²+1

g(x)= (x – 1)² + 1

The complete question is

Consider the function f(x) = (x + 2)2 + 1. Which of the following functions shifts the graph of f(x) to the right three units?

g(x) = (x + 5)2 + 1

g(x) = (x + 2)2 + 3

g(x) = (x – 1)2 + 1

g(x) = (x + 2)2 – 2

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0.2342 is the probability that the number of women that would prefer to drink tea over coffee is within 1 standard deviation of the mean

Let's start by finding the mean and standard deviation of the number of women who prefer to drink tea over coffee in a sample of 7 women.

The mean is given by:

μ = np

where n is the sample size and p is the probability of success (the proportion of women who prefer tea over coffee).

So, for this problem, μ = 7 * 0.31 = 2.17

The standard deviation is given by:

σ = sqrt(np(1-p))

where sqrt denotes the square root.

So, for this problem, σ = sqrt(7 * 0.31 * (1 - 0.31)) = 1.22

2.17 - 1.22, 2.17 + 1.22

= 0.95, 3.39

using binomial probability

p^x * 1 - p

= 0.31 * 7 x 0.108

= 0.2342

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Answer: -4x(x-1)

Step-by-step explanation:

Since both are perfect squares just factor.

a) The percentage of residents who liked the local parks out of those surveyed is 30%.

b) The percentage of the residents who liked the school system out of those surveyed is 60%.

The percentage refers to a portion of a whole value or quantity, expressed in percentage terms.

The percentage is a ratio, which compares a value of interest with the whole, and is computed by multiplying the quotient of the division operation between the particular value and the whole value by 100.

The total number of Plana residents surveyed = 240

The number of residents who responded that they liked the local parks = 72

The percentage of residents who liked the local parks = 30% (72 ÷ 240 x 100)

The number of residents who responded that they liked the school system = 144

The percentage of residents who liked the school system = 60% (144 ÷ 240 x 100).

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the first term of the geometric series is approximately 105.294.

How to solve the problem?

The formula for the sum of a geometric series is given by:

Sn = a1(1 - rⁿ)/(1 - r)

where a1 is the first term, r is the common ratio, n is the number of terms, and Sn is the sum of the series.

We are given that Sn = 3045, r = 2/5, and An = 120, where An is the nth term of the series. We can use these values to solve for a1.

First, we can use the formula for the nth term of a geometric series:

An = a1 * rⁿ⁻¹

Substituting An = 120 and r = 2/5, we get:

120 = a1 * (2/5)ⁿ⁻¹

We also know that the sum of the series is Sn = 3045. Substituting the formula for Sn and solving for n, we get:

3045 = a1(1 - (2/5)ⁿ)/(1 - 2/5)

12180 - 4(2/5)ⁿ = 5a1

a1 = (12180 - 4(2/5)ⁿ)/5

Now we can substitute this expression for a1 into the equation we obtained for An:

120 = [(12180 - 4(2/5)ⁿ)/5] * (2/5)ⁿ⁻¹

Simplifying and solving for n, we get:

n = log(12/5) / log(2/5)

Substituting this value of n back into the expression for a1, we get:

a1 = (12180 - 4(2/5) power (log(12//log(2/5))) / 5

Using a calculator, we can evaluate this expression to get:

a1 ≈ 105.294

Therefore, the first term of the geometric series is approximately 105.294.

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Answer:

54 degrees

Step-by-step explanation:

Any angle subtended by the angle is twice the angle subtended by the same angle.

Or m<LMP =2m<LNM.

So m<LNM = 108/2 = 54 degrees.

Answer:

To calculate the monthly payment for each option, we can use the loan formula:

Payment = (P * r) / (1 - (1 + r)^(-n))

where P is the principal amount, r is the monthly interest rate, and n is the total number of payments.

For Option A, the principal amount is $60,000, the interest rate is 4% per year, and the loan term is 10 years. We first need to convert the annual interest rate to a monthly interest rate:

r = 4% / 12 = 0.00333333 (rounded to 8 decimal places)

n = 10 years * 12 months/year = 120 months

Using the loan formula, we get:

Payment = (60000 * 0.00333333) / (1 - (1 + 0.00333333)^(-120)) = $630.55

Therefore, the monthly payment for Option A is $630.55.

For Option B, the principal amount is also $60,000, the interest rate is 3% per year, and the loan term is 20 years. We convert the annual interest rate to a monthly interest rate:

r = 3% / 12 = 0.0025 (rounded to 4 decimal places)

n = 20 years * 12 months/year = 240 months

Using the loan formula, we get:

Payment = (60000 * 0.0025) / (1 - (1 + 0.0025)^(-240)) = $342.61

Therefore, the monthly payment for Option B is $342.61.

To calculate the total amount paid over the life of the loan for each option, we simply multiply the monthly payment by the total number of payments:

For Option A, the total amount paid = $630.55 * 120 months = $75,665.92

For Option B, the total amount paid = $342.61 * 240 months = $82,226.40

To calculate the total interest paid over the life of the loan for each option, we subtract the principal amount from the total amount paid:

For Option A, the total interest paid = $75,665.92 - $60,000 = $15,665.92

For Option B, the total interest paid = $82,226.40 - $60,000 = $22,226.40

Therefore, Option A has a lower monthly payment and total amount paid over the life of the loan, but Option B has a longer loan term and a lower interest rate, resulting in a higher total interest paid over the life of the loan

After answering the presented question, we can conclude that inequality therefore, the solution for z is z < -7.

In mathematics, an inequality is a non-equal connection between two expressions or values. As a result, imbalance leads to inequity. In mathematics, an inequality connects two values that are not equal. Inequality is not the same as equality. When two values are not equal, the not equal symbol is typically used (). Various disparities, no matter how little or huge, are utilised to contrast values. Many simple inequalities can be solved by altering the two sides until just the variables remain. Yet, a lot of factors contribute to inequality: Negative values are divided or added on both sides. Exchange left and right.

[tex]15 - 3(2 - z) < -12\\15 - 6 + 3z < -12 \\9 + 3z < -12 \\3z < -21 \\z < -7 \\[/tex]

Therefore, the solution for z is z < -7.

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Answer:

The circumcenter of the triangle with vertices (-7, -1), (-1, -1), and (-7, -9) is (-4, -8).

Step-by-step explanation:

To find the coordinates of the circumcenter of the triangle with vertices (-7, -1), (-1, -1), and (-7, -9), we can use the following steps:

Step 1: Find the midpoint of two sides

We first find the midpoint of two sides of the triangle. Let's take sides AB and BC:

Midpoint of AB: ((-7 + (-1))/2, (-1 + (-1))/2) = (-4, -1)

Midpoint of BC: ((-1 + (-7))/2, (-1 + (-9))/2) = (-4, -5)

Step 2: Find the slope of two sides

Next, we find the slope of the two sides AB and BC:

Slope of AB: (-1 - (-1))/(-1 - (-7)) = 0/6 = 0

Slope of BC: (-9 - (-1))/(-7 - (-1)) = -8/(-6) = 4/3

Step 3: Find the perpendicular bisectors of two sides

We can now find the equations of the perpendicular bisectors of the two sides AB and BC. Since the slope of the perpendicular bisector is the negative reciprocal of the slope of the side, we have:

Equation of perpendicular bisector of AB:

y - (-1) = (1/0)[x - (-4)]

x = -4

Equation of perpendicular bisector of BC:

y - (-5) = (-3/4)[x - (-4)]

y + 5 = (-3/4)x - 3

y = (-3/4)x - 8

Step 4: Find the intersection of perpendicular bisectors

We now find the point of intersection of the two perpendicular bisectors. Solving for x and y from the two equations, we get:

(-4, -8)

Therefore, the circumcenter of the triangle with vertices (-7, -1), (-1, -1), and (-7, -9) is (-4, -8).

The distance between point A (6,3) and A' (2,1) is half of the original distance between A and B (6,3) and (15,3). Therefore, the scale factor is one half.

To determine the scale factor used to create the image of the second polygon A'B'C'D' from the original polygon ABCD, we can compare the side lengths of the two polygons.

The scale factor used to create the image is one half. This is because the distance between each vertex in the original polygon and the corresponding vertex in the image polygon is half of the original distance. For example,
Let's take the horizontal sides:
Side AB in polygon ABCD has length 15 - 6 = 9 units.
Side A'B' in polygon A'B'C'D' has length 5 - 2 = 3 units.

Now, we can find the scale factor by dividing the side length of A'B' by the side length of AB:
Scale factor = (A'B') / (AB) = 3 / 9 = 1/3

So, the scale factor used to create the image is one third.

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