For questions 4-10, Circles C and M are shown. Lines PL and GK intersect at point Z. Line PL is tangent to circle C at point P and circle M at point L. Line GK is tangent to circle c at point G and circle M at point L. PZ = 10y - 3, ZL = 4x + 10, GZ = 7y + 21, ZK = 6x - 16.

4:Write an equation you can use to solve for x. 5: Solve the equation you wrote in question 4 for x.
6. Write an equation you can use to solve for y.
7: Solve the equation you wrote for 6 for y.
8: What is the length of GK?
9. What is the length of GZ?
10. What is the length of ZL?​

Answer:

D: 27(12x² - 5x - 2).

Step-by-step explanation:

The formula for the surface area of a cylinder is: A = 2πr² + 2πrh

Given that the radius of the cylinder is 3x - 2 cm and the height is x + 3 cm, we can substitute these values in the formula and simplify:

A = 2π(3x - 2)² + 2π(3x - 2)(x + 3)

A = 2π(9x² - 12x + 4) + 2π(3x² + 7x - 6)

A = 18πx² - 24πx + 8π + 6πx² + 14πx - 12π

A = 24πx² - 10πx - 4π

A = 2π(12x² - 5x - 2)

Therefore, the answer is option D: 27(12x² - 5x - 2).

Answer:

D

Step-by-step explanation:

Answer: Circumference = 18 π cm

Step-by-step explanation:

We can use this formula to find the circumference of a circle:

➜ r is equal to the radius, also known as half the width of a circle.

      C = 2πr

We will substitute our known values and solve for C by multiplying. Since the answer option includes π in our units, we do not multiply this into the 18.

      C = 2πr

      C = 2π(9)

      C = 18 π cm

Answer: 18π

Step-by-step explanation:

The circumference of a circle is equal to 2πr, with r being equal to the radius. The radius of a circle is defined as the length from any point on the circle to its middlemost point. In this case, we are given the value of the radius as defined by the line. The radius equals 9, which we can plug into the equation.

C= 2πr

C = 2π(9)

C = 18π = 56.55

Hope this helps!

To ensure that smooth curves have no cusps, we need to require that z'(t) never vanishes. This condition ensures that the tangent line to the curve changes smoothly and continuously as we move along the curve, without any abrupt changes in direction that would create cusps.

To understand why the condition that z'(t) never vanishes is necessary to ensure that smooth curves have no cusps, we first need to understand what a cusp is. A cusp is a point on a curve where the tangent line changes direction abruptly, creating a sharp point or corner in the curve.

Now, let's consider the curve traced by the parametrization z(t) = t^2it^3 with -1 ≤ t ≤ 1. To determine whether this curve has any cusps, we need to calculate the derivative of z(t) with respect to t:

z'(t) = 2it^3 + 3t^2i

If we set z'(t) equal to zero and solve for t, we get:

2it^3 + 3t^2i = 0
t^2(2i t + 3i) = 0

This equation has two solutions: t = 0 and t = -3/2i. These are the points on the curve where z'(t) vanishes.

At t = 0, the curve passes through the origin, which is a smooth point. However, at t = -3/2i, the curve has a cusp. To see why, we can look at the behavior of z(t) near this point.

As t approaches -3/2i from either side, the magnitude of t^2 increases without bound, while the magnitude of t^3 remains constant. This means that z(t) approaches infinity along a straight line with slope -3/2i. In other words, the curve has a sharp corner or cusp at this point.

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Sample mean in sample b to be less variable and more representative of  population mean than sample mean in sample a.

Standard error measures the variability of sample means around the population mean.

It decreases as sample size increases.

Using the formula for the standard error of the mean,

Calculate the standard errors for each sample,

Standard error of sample a

= s/√(n)

= 850/√(9)

= 283.33

Standard error of sample b

= s/√(n)

= 850/√(36)

= 141.67

Where s is the sample standard deviation

And n is the sample size.

The standard error of sample b is smaller than the standard error of sample a.

Sample mean in sample b is more likely to be closer to true population mean than sample mean in sample a.

Therefore, sample mean b is more surprising as larger sample size makes it more likely and accurately represents population mean.

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A. Talia can use 1 and 3/4 cup strawberries to double the recipe. B. Talia can use 1 and 1/3 cup yogurt to make 3 servings. D. Talia can use 2 cups orange juice to make 4 servings.

In statistics, a proportion is a fraction or a percentage that reflects the proportion of a population's or sample's members who share a particular attribute to the population's or sample's overall size. It is a kind of ratio where the denominator is the entire population or sample size and the numerator is the number of people who possess a specific trait or attribute.

From the given ingredients and the serving size we can see that the correct statement is:

A. Talia can use 1 and 3/4 cup strawberries to double the recipe.

B. Talia can use 1 and 1/3 cup yogurt to make 3 servings.

D. Talia can use 2 cups orange juice to make 4 servings.

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The complete question is:

The average distance of electric cars is 275.

The range of the data set is 300.

Given:

To find: Average distance of electric cars

Solution:

The average distance of electric cars can be calculated by finding the average of the data set containing the distances of electric cars. Let's assume that the data set is as follows:

250, 250, 250, 400, 300, 300, 350, 100

To find the average distance of electric cars, we can use the formula:

Average = (Sum of all the data points) / (Number of data points)

Substituting the given values, we get:

Average = (250 + 250 + 250 + 400 + 300 + 300 + 350 + 100) / 8

Average = 2200 / 8

Average = 275

Therefore, the average distance of electric cars is 275.

Now, let's calculate the range of the data set. The range is the difference between the maximum and minimum values in the data set. From the given data set, we can see that the minimum value is 100 and the maximum value is 400.

Range = Maximum value - Minimum value

Range = 400 - 100

Range = 300

Therefore, the range of the data set is 300.

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The scale factor of the dilation must be greater than 1. if the image A'B'C' is bigger than the preimage ABC

If the image A'B'C' is bigger than the preimage ABC, then the scale factor of the dilation must be greater than 1.

In other words, the length of any segment in the image is larger than the corresponding segment in the preimage by the same factor.

Since D is the center of the dilation, we can use the ratio of the corresponding side lengths of the image and preimage triangles to find the scale factor.

For example, if we want to find the scale factor for A', we can use:

scale factor = B'C'/BC

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The volume of the prism is gotten as 312 cm³.

Volume is described as a measure of three-dimensional space. It is often quantified numerically using SI derived units or by various imperial or US customary units.

The basic formula for volume is length × width × height.

The formula for the volume of triangular prism = area of triangle x length

area of triangle = 1/2 x base x height

⇒ area of triangle = 1/2 x 8 x 6 = 24 cm²

Therefore, volume of the prism = 24 x 13 = 312 cm³

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Answer:

Step-by-step explanation:

Graph #          Matching equation

1                             |-3x|

2                            |-x|

3                           -|2x|

You can tell which one matches by finding the slope and whether the V points up or down

The rate at which the base of the triangle is changing when the altitude is 20 cm and the area is 120 cm^2 is 0 cm/s.

To solve this problem, we need to use the formula for the area of a triangle, which is:

A = (1/2)bh

Where A is the area, b is the base, and h is the altitude.

We know that the area is 120 cm^2 and the altitude is 20 cm. So we can plug in these values and solve for the base:

120 = (1/2)b(20)

240 = 20b

b = 12 cm

Now we need to differentiate both sides of the equation with respect to time (t):

dA/dt = (1/2)(db/dt)h

We are given the value of dh/dt (which represents the rate at which the altitude is changing) is 3 cm/s. We need to find db/dt (which represents the rate at which the base is changing).

Plugging in the values we know, we get:

dA/dt = (1/2)(db/dt)(20)

Solving for db/dt, we get:

db/dt = (2dA/dt)/h

Plugging in the values we know, we get:

db/dt = (2)(0)/20

db/dt = 0 cm/s
Since the area is constant (120 cm²) and the altitude is constant (20 cm), the base of the triangle is not changing. Therefore, the rate at which the base is changing is: 0 cm/s

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Thus, the number of current lawnmowers in the Springtown Hardware is 864,900.

In mathematics, a percentage is a number or ratio that may be expressed as a fraction of 100. The Latin word "per centum," which meaning "per 100," is where the word "percent" comes from. % is the symbol used to represent percentages.

When a number is expressed in decimal form, you can calculate its percentage by multiplying it by 100. For instance, multiplying 0.5 by 100 gives you the percentage 50%.

Given data:

Springtown Hardware's inventory - 697,500 lawnmowers.

Current inventory of 24% more lawnmowers.

Current inventory = old inventory  + 24% of old inventory

Current inventory =  697,500 + 24 % of  697,500

Current inventory =  697,500 + 24 *  697,500/ 100

Current inventory =  697,500 + 24* 6,975

Current inventory = 864,900

Thus, the number of current lawnmowers in the Springtown Hardware is 864,900.

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I'm sorry, but I cannot provide a definitive answer to your question as I do not have access to the table you are referring to.

However, in general, to find the price per spool of each kind of thread, you would need to know the total price of the package and the number of spools in the package.

To calculate the price per spool, you would divide the total price of the package by the number of spools in the package. For example, if a package costs $10 and contains 50 spools of thread, the price per spool would be $0.20 ($10 ÷ 50 = $0.20).

You can use this method to find the price per spool for each type of thread in the table you are looking at.

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p=8h is the equation.This part is represented in tabular form.The coefficient of h. Mark worked 21 hours that week.Yes, the equation is a direct variation.

An equation is a statement that shows the equality of two expressions. It typically contains one or more variables and may involve mathematical operations such as addition, subtraction, multiplication, and division. Equations are commonly used in mathematics, science, and engineering to model real-world situations and solve problems.

Direct variation is a mathematical relationship between two variables, where a change in one variable results in a proportional change in the other variable. In other words, if two variables are in direct variation, when one variable increases, the other variable increases as well, and when one variable decreases, the other variable decreases as well, in a constant ratio.

Part A: The equation for this situation is p = 8h, where p represents the pay in dollars and h represents the number of hours worked.

Part B:

| Worked (hrs) | Pay (p) |

|------------------|---------|

| 1                | 8       |

| 2                | 16      |

| 3                | 24      |

| 4                | 32      |

| 5                | 40      |

| 6                | 48      |

and so on.

Part C: The coefficient of h, which is 8, shows the number of hours Mark worked.

Part D: We can use the equation p = 8h and substitute p = 168 to find the number of hours Mark worked.

168 = 8h

h = 21

Therefore, Mark worked 21 hours that week.

Part E: Yes, the equation is a direct variation because the pay (p) is directly proportional to the number of hours worked (h) and the constant of variation is 8.

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The solution to the IVP given by y''+y=t, y(0)=1, y'(0)=-2 is y(t) = cos(t) - (3/2) sin(t) + (1/2) t.

To solve the Initial Value Problem, we can use the method of undetermined coefficients, which involves assuming a particular form for the solution to the non-homogeneous equation y'' + y = t, and then finding the coefficients of the terms in that form by substituting it back into the equation.

First, we find the general solution to the homogeneous equation y'' + y = 0

The characteristic equation is r² + 1 = 0, which has solutions r = ±i. Therefore, the general solution to the homogeneous equation is

y_h(t) = c₁ cos(t) + c₂ sin(t),

where c₁ and c₂ are constants determined by the initial conditions.

Next, we assume a particular form for the non-homogeneous solution, based on the form of the right-hand side t. Since t is a linear function, we assume that the particular solution has the form

y_p(t) = a t + b.

Substituting this into the differential equation, we get

y''_p + y_p = t

2a + (at+b) = t.

Equating coefficients, we get

a = 1/2, b = 0.

Therefore, the particular solution is

y_p(t) = (1/2) t.

The general solution to the non-homogeneous equation is then the sum of the homogeneous and particular solutions

y(t) = y_h(t) + y_p(t)

= c₁ cos(t) + c₂ sin(t) + (1/2) t.

To determine the constants c₁ and c₂, we use the initial conditions:

y(0) = c₁ cos(0) + c₂ sin(0) + (1/2) (0) = c₁ = 1,

y'(0) = -c₁ sin(0) + c₂ cos(0) + (1/2) (1) = c₂ - (1/2) = -2,

so c₂ = -3/2.

Therefore, the solution to the IVP is

y(t) = cos(t) - (3/2) sin(t) + (1/2) t.

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Answer:

Harry can buy 5 gallons of lemonade and 5 gallons of iced tea and still have $1.50 left over from his original $24 dollar budget

Step-by-step explanation:

The theorem is shown in a pythagoras theorem is c² = a² + b²

The theorem shown in the Pythagorean theorem is "a² + b² = c²". This theorem is named after the ancient Greek mathematician Pythagoras, who is credited with discovering it.

The Pythagorean theorem applies to right-angled triangles and states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Mathematically, we can express this as:

c² = a² + b²

Where c is the length of the hypotenuse, and a and b are the lengths of the other two sides (called the legs) of the right-angled triangle.

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The probability is 0.9835 where the sample mean would be greater than 59.5 kilograms.

The central limit theorem can be used to approximate the sampling distribution of the sample mean as a normal distribution.

where the mean equals the population mean (μ = 62 kg) and the standard deviation equals the population standard deviation divided by the square root of sample size (σ/sqrt (n) = sqrt(144 / 195) = 2.4287 kg).

Next, find the probability that the sample mean is greater than 59.5 kilograms.

z = (59.5 - 62) / (2.4287 / square (195)) = -2.1295

Using an ordinary regular table or calculator, we know that the probability of a Z-score less than -2.1295 is 0.0165.

Therefore, the probability that the sample mean exceeds 59.5 kilograms is

1 - 0.0165 = 0.9835

Rounding to four decimal places gives a probability of 0.9835.

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The value of x for the given triangle through which the perimeter of the given relation is satisfied is 10.

What about perimeter of triangle?

The perimeter of a triangle is the total length of its boundary, which is the sum of the lengths of its three sides. The perimeter can be thought of as the distance around the triangle, and it is measured in units of length such as centimeters, meters, or feet. The perimeter of a triangle is an important geometric property that is used in many practical applications, such as calculating the amount of fencing needed to enclose a triangular-shaped garden or determining the length of wire required to form a triangular circuit.

According to the given information:

The perimeter of triangle is sum of all sides of the triangle

In which,

2x + 4 + 3x - 8 + x - 2 = 54

6x - 6  = 54

6x = 60

x = 10

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Answer:

Yo wassup bro, at this computer factory, they be making two types of computers, laptops and desktops. They churn out 70 laptops a day and 55 desktops a day. They got a total of 14 machines to make these computers. And they make a total of 905 computers a day. We gotta figure out how many machines are making laptops and how many are making desktops, ya know?

Alright, let's set up a system of equations to solve this. Let's call the number of machines making laptops "x" and the number of machines making desktops "y".

So, we got two equations here:

The total number of computers they make in a day is 905, so we can write: x laptops + y desktops = 905.

They got a total of 14 machines, so we can write: x + y = 14.

Now, let's solve this system of equations to find the values of x and y, man. Once we got those, we'll know how many machines are making laptops and how many are making desktops at this computer factory, yo!

The volume of water in the tank won't be enough for the fish.

A cuboid is a solid shape or a three-dimensional shape. The volume of a cuboid is expressed as;

V = l×w× h

where l is the length, w is the width and h is the height.

The volume of water needed by the fish is 700,000× 1litre = 700,000 litres.

The volume of water the tank can occupy is

V = l× b× h

V = 80× 1200× 70

V = 6,720,000cm³

in litres ;

V = 6,720,000/1000

V = 6720litres

therefore the volume water in the tank is not enough for the fish

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Thus, the correct Percent response is B) 79%.

Percentage refers to a portion of every hundred. Although the abbreviations "pct.", "pct.", and occasionally "pc" are also used, it is frequently shown using the percent sign, "%".

If you have 100 apples and distribute 10 of them, for instance, you have distributed 10% of your total apple supply.

We deduct Ms. Yamato's deductions from her gross income to determine her take-home pay:

$2644 - $548.30 = $2095.702.

By dividing her take-home pay by her gross pay and multiplying the result by 100%, we can determine what proportion of her gross pay is taken home:

($2095.70 / $2644) * 100% = 79%

Thus, the correct response is B) 79%.

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If the  volume of a volleyball is approximately 113 in³, then the diameter of the volleyball is approximately option (D) 5 inches.

The formula for the volume of a sphere is

V = (4/3)πr³

where V is the volume, π is pi (approximately 3.14), and r is the radius of the sphere.

We are given the volume of the volleyball as 113 in³, so we can solve for the radius as follows

113 = (4/3)πr³

r³ = 113 / ((4/3)π)

r³ ≈ 21.50

r ≈ 2.8

To find the diameter, we double the radius:

d = 2r ≈ 5.6

Rounding this to the nearest whole number

d = 5 in

Therefore, the correct option is (D) 5 in

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Answer:

Surface area of prism A is approximately 12.4 square units.

Step-by-step explanation:

Since prism A and prism B are similar, their corresponding side lengths are proportional. Let's use the scale factor k to represent the ratio of the side lengths of prism B to those of prism A. Then, the volume of prism B is (k³)(27) = 27k³ cubic units. Similarly, the surface area of prism B is (k²)(surface area of prism A).

We are given that the volume of prism B is approximately 512 cubic units. Therefore, we can solve for k:

27k³ = 512

k³ = 512/27

k ≈ 3.62

Now, we can use k to find the surface area of prism A:

surface area of prism B = (k²)(surface area of prism A)

128 = (3.62²)(surface area of prism A)

surface area of prism A ≈ 12.4 square units

Therefore, the surface area of prism A is approximately 12.4 square units.

Answer:

[tex]\frac{24\sqrt{5} }{5}[/tex]

Step-by-step explanation:

Let's start by assigning one of the unknown legs with the variable x.

We know that the other leg is 2 times the length of x, so we can write:

2x

We also know that the length of the hypotenuse is 12.

From here, we can use the Pythagorean Theorem.

Recall that the Pythagorean Theorem is:

[tex]a^2+b^2=c^2[/tex]

where a is the length of one leg, b is the length of the other leg, and c is the length of the hypotenuse.

Let's substitute the values. We have:

[tex]x^2+(2x^2)=12^2=\\x^2+4x^2=144=\\5x^2=144=\\x^2=\frac{144}{5}=\\x=\frac{12}{\sqrt{5} }[/tex]

Let's rationalize the denominator by multiplying the numerator and denominator by [tex]\sqrt{5}[/tex], like so:

[tex]\frac{12}{\sqrt{5} } =\\\frac{12\sqrt{5} }{5}[/tex]

Therefore, [tex]x=\frac{12\sqrt{5} }{5}[/tex]

Let's solve for 2x:

[tex]2x=\\2(\frac{12\sqrt{5} }{5})=\\ \frac{24\sqrt{5} }{5}[/tex]

So, the length of the longest leg is [tex]\frac{24\sqrt{5} }{5}[/tex]

the sequence of transformations that produces R'S'T' from RST is translation 2 units to the right and 3 units down, followed by a rotation of 90 degrees clockwise, and finally a reflection across the x-axis.

A triangle is a polygon since it has a total of three faces along with three vertices. It belongs to the fundamental geometric shapes. Triangles having vertex positions alternating between A and B are referred to as triangles ABC. A unique plane and a triangle in Euclidean geometry appear when the three vertices are not collinear. Due to its three sides and three corners, triangles are considered polygons. The corners of a triangle are defined as the locations where its three sides meet. Three triangle angles are multiplied to get 180 degrees.

To transform triangle RST to R'S'T', we can use the following sequence of transformations:

Translation: Move the triangle 2 units to the right and 3 units down. This will result in triangle R''S''T'' with vertices (2, -3), (-2, 0), (-3, -2).

Rotation: Rotate triangle R''S''T'' 90 degrees clockwise about the origin. This will result in triangle R'''S'''T''' with vertices (3, 2), (0, 2), (2, 3).

Reflection: Reflect triangle R'''S'''T''' across the x-axis. This will result in triangle R'S'T' with vertices (3, -2), (0, -2), (2, -3).

Therefore, the sequence of transformations that produces R'S'T' from RST is translation 2 units to the right and 3 units down, followed by a rotation of 90 degrees clockwise, and finally a reflection across the x-axis.

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The period of a cosine function of the form y = cos bx is equal to T= 2π/b where 'T' is the period of the cosine function and b is the coefficient of x in the function.

This formula tells us that the period of the cosine function is equal to the length of one complete cycle of the function.

it represents  the distance along the x-axis for the cosine function to complete one full oscillation.

The period of a cosine function of the form y = cos bx, first identify the coefficient b.

Use the formula T= 2π/b  to calculate the period 'T'.

For example,

Consider cosine function y = cos 2x,

The coefficient of x is 2.

Using the formula above, the period is equal to

Period = 2π/2

           = π

So the period of the function y = cos 2x is π.

This implies that the cosine function completes one full oscillation every π units along the x-axis.

Therefore, the formula used to calculate the period of the cosine function y = cos bx is given by T= 2π/b.

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the 11th term in the given geometric sequence is equal to [tex]-9216x^{52}[/tex]

A sequence is a list of numbers indexed by the natural numbers 1,2,3,4,5.. A common notation for a sequence is [tex]a_1,a_2,a_3,\cdots[/tex]. Most often for interesting sequences there is some pattern in the list of numbers, and there is some formula to generate the nth number in the sequence. A sequence is called a geometric sequence if the next number number is got from the previous number by multiplying it by a fixed number r. This number is called the common ratio r of the sequence. So if the first number is a, then the second number is ar, the third is [tex]ar^2[/tex]... So that the nth number [tex]a_n = ar^{n-1][/tex].  

For example 1 ,3,9,27,81,.... The sum of the first n terms of such a sequence is equal to [tex]$S_n = \frac{a(r^n-1)}{r-1}$[/tex]. if the absolute value of r, [tex]|r| < 1[/tex], then the sum to infinity is [tex]$S = \frac{a}{1-r}$[/tex].

In our question we are given a geometric sequence whose first term :

[tex]$a = -9x^2\,,\textrm{ and common ratio : r } = 2x^5$[/tex] .

So the 11th term is [tex]$ar^{10} = (-9x^2){(2x^5)}^{10} = (-9x^2)(1024x^{50}) = -9216x^{52}$[/tex]

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Answer: Trapezoid

Step-by-step explanation:

Answer:

it's also called a quadrilateral with 1 right angle

The correct option is C′(−10.5, 3.5) i.e. the vertex of point C′ is (-10.5, 3.5).

What is dilation?

In mathematics, dilation is a type of transformation that changes the size of a geometric object, but not its shape or orientation. Dilation is similar to scaling, but it involves a fixed point called the center of dilation, and a scale factor that determines the degree of expansion or contraction.

To determine the vertex of point C′, we need to apply the dilation transformation to the coordinates of the original vertex C(-3,1) using a scale factor of 3.5.

The coordinates of C′ can be found by multiplying the coordinates of C by the scale factor:

C′ = (3.5)C = (3.5)(-3,1) = (-10.5,3.5)

Therefore, the vertex of point C′ is (-10.5, 3.5).

Thus, the correct option is C′(−10.5, 3.5).

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For the image with coordinates A'(3, 5), B'(3, -1) and C'(-2, 0) occurred after the x-axis refection.

The x-coordinate remains constant when a point is reflected across the x-axis, but the y-coordinate is assumed to be the additive inverse. Point (x, y) is reflected across the x-axis as (x, -y).

The y-coordinate stays the same when a point is reflected across the y-axis, but the x-coordinate is assumed to be the additive inverse. Point (x, y) is reflected across the y-axis as (-x, y).

Given data:

pre-image coordinates - A(3, -5), B(3, 1) and C(-2, 0).

Image coordinates A'(3, 5), B'(3, -1) and C'(-2, 0).

As its is clear from the given coordinates that value of x is same while the value of y gets change by negative.

(x,y) ---> (x, -y) , shows the refection about the x axis.

Thus, For the image with coordinates A'(3, 5), B'(3, -1) and C'(-2, 0) occurred after the refection about the x axis.

Know ore about the reflection about x axis:

https://brainly.com/question/24696463

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