PROOFS please fill in blanks need help badly !! would gratefully appreciate it
9. Given: U is the midpoint of ST, SVTW, VU =WU
Prove: SVU = ZTWU

The answer is option D: Reflect trapezoid EFGH across side HG, translate it three units up and four units to the right, and dilate it by a scale factor of 1/2.

In mathematics, a scale factor is a ratio that describes how the size or dimensions of an object change when it is scaled up or down. It is the ratio of the size or measurement of an original object to the size or measurement of a new object that has been scaled by a certain factor.

For example, if the length of an object is multiplied by a scale factor of 2, the new object will be twice as long as the original object. Similarly, if the area of an object is multiplied by a scale factor of 3, the new object will have three times the area of the original object.

To show that trapezoid E’F’G’H is similar to trapezoid EFGH, we need to apply a sequence of transformations that preserves shape and size.

First, we notice that trapezoid EFGH can be transformed into trapezoid E’F’G’H by a dilation centered at the origin with a scale factor of 1/4, followed by a translation 3 units to the right and 4 units up. So, we need to find the transformation that maps EFGH onto E’F’G’H.

Option B reflects trapezoid EFGH across side EF, translates it 5 units up and 8 units to the right, and dilates it by a scale factor of 1/2. This transformation does not preserve shape and size, so it cannot be the correct option.

Option A reflects trapezoid EFGH across side HG, translates it 8 units up and 5 units to the right, and dilates it by a scale factor of 2. This transformation does not preserve shape and size either, so it cannot be the correct option.

Option C reflects trapezoid EFGH across side EF, translates it 4 units up and 3 units to the right, and dilates it by a scale factor of 2. This transformation does not preserve shape and size, so it cannot be the correct option.

Option D reflects trapezoid EFGH across side HG, translates it 3 units up and 4 units to the right, and dilates it by a scale factor of 1/2. This transformation preserves shape and size, so it is the correct option.

Therefore, the answer is option D: Reflect trapezoid EFGH across side HG, translate it three units up and four units to the right, and dilate it by a scale factor of 1/2.

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Answer:

Step-by-step explanation:

y = x² -10x + 16

y = (x - 8) (x - 2)

solve each when y = 0

x - 8 = 0

x - 8 + 8 = 0 +8

x = 8

and

x - 2 = 0

x - 2 + 2 = 0 +2

x = 2

SO solutions are (8, 0) and (2, 0)

In order to find the area of the given rectangle, we can use square-inch tiles or rulers. the area of the rectangle is 30 square inches. The length of the rectangle is 15 inches, width is 2 inches then  area of 30 square inches.

Using square-inch tiles:

We can cover the rectangle with square-inch tiles and count the total number of tiles to find the area. Each tile represents one square inch, and the area of the rectangle is equal to the number of tiles covering it. We can start by covering one side of the rectangle with tiles, and then count the number of tiles used. The length of the rectangle is 15 inches, so we need to use 15 tiles to cover one side. We can then cover the other side with the same number of tiles, which gives us a total of 30 tiles. Therefore, the area of the rectangle is 30 square inches.

Using a ruler:

We can measure the length and width of the rectangle using a ruler and then multiply the two measurements to find the area. The length of the rectangle is 15 inches, and the width is 2 inches. Multiplying these values gives us an area of 30 square inches.

Using circular counters:

Circular counters are not an appropriate tool to find the area of a rectangle because they are circular in shape and cannot cover the entire surface of the rectangle.

In conclusion, to find the area of a rectangle, we can use appropriate tools such as square-inch tiles or rulers. These tools allow us to accurately measure the length and width of the rectangle and calculate the area by counting or multiplying the measurements.

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The probability that the 39 randomly-selected laptops have a mean "replacement-time" as 3.8 years or less is equal to 0.1058.

The distribution of the sample mean follows a normal distribution with mean (μ) = 3.9 years and standard deviation = σ/√n = 0.5/√39 years, where n = 39 is the sample size.

We want to find the probability that the sample mean replacement time is less than or equal to 3.8 years, which is written as : P(x' ≤ 3.8).

By using the z-score formula, we convert this to a standard normal distribution:

⇒ z = (x' - μ)/(σ/√n) = (3.8 - 3.9)/(0.5/√39) = -1.249,

Using a standard normal distribution table, the probability that a standard normal variable is less than or equal to -1.249,is approximately 0.1058,

Therefore, the required probability is approximately 0.1058.

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The given question is incomplete, the complete question is

Assuming that the laptop replacement times have a mean of 3.9 years and a standard deviation of 0.5 years, find the probability that 39 randomly selected laptops will have a mean replacement time of 3.8 years or less.

a. The probability that a randomly selected pipe is made of PVC is approximately 0.242 or 24.2%.

b. The probability that randomly selected pipe is a trunk is 0.104

c. The probability that randomly selected pipe is a main made of galvanized iron (gi) is 0

d. The probability that randomly selected pipe is  a submain made of asbestos cement is 0.168

What is probability?

The chance of an event can be calculated using the probability formula by simply dividing the favorable number of possibilities by the total number of outcomes. The likelihood of an event occurring can be anything between 0 and 1, as the favorable number of outcomes can never exceed the total number of outcomes.

a. The total number of pipes is 8646, and the number of pipes made of PVC is 2092. The probability of a randomly selected pipe being made of PVC is:

P(PVC) = 2092/8646 ≈ 0.242

b. The total number of pipes is 8646, and the number of trunk pipes is 903. The probability of a randomly selected pipe being a trunk is:

P(trunk) = 903/8646 ≈ 0.104

c. The total number of main pipes is 5299, and the number of main pipes made of galvanized iron (GI) is 0 (according to the table). Therefore, the probability of a randomly selected main pipe being made of GI is:

P(GI|main) = 0/5299 = 0

Note that the vertical bar "|" means "given" or "conditioned on".

d. The total number of submain pipes is 2444, and the number of submain pipes made of asbestos cement (AC) is 411. Therefore, the probability of a randomly selected submain pipe being made of AC is:

P(AC|submain) = 411/2444 ≈ 0.168

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In conclusion Shaniyah has $159.25 for food and gas.

How to find?

First, we need to find how much Shaniyah spends on internet service and cell phone bill:

Amount spent on internet service = 21% of $325 = 0.21 x $325 = $68.25

Amount spent on cell phone bill = 3/10 of $325 = (3/10) x $325 = $97.50

The total amount spent on internet service and cell phone bill is $68.25 + $97.50 = $165.75

To find how much Shaniyah has left for food and gas, we subtract the total amount spent on internet service and cell phone bill from the total amount spent on bills:

Amount spent on food and gas = Total amount spent on bills - Amount spent on internet service and cell phone bill

Amount spent on food and gas = $325 - $165.75

Amount spent on food and gas = $159.25

Therefore, Shaniyah has $159.25 for food and gas.

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The proportions from the sets of numbers are 66/11 = 6/1, 35/14 = 25/10, 48/12 = 4/1, 45/9 = 25/5 and 72/12 = 6/1

To create a proportion from each set of numbers, we must select 4 numbers from each set and express them as a fraction in the form of a/b = c/d.

Here are the proportions for each set:

Note: To create a proportion, we must select numbers that are related to each other in some way, such as the sides of a shape or the parts of a ratio.

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Answer:

Event 1 and Event 2 have different probabilities, and it is not possible to determine which one is higher without additional information.

Event 1 involves rolling a six-sided dice, which has six possible outcomes, each with an equal probability of 1/6. The probability of rolling a 3 on a single roll is 1/6. Therefore, the probability of rolling two 3's in a row is (1/6) x (1/6) = 1/36.

Event 2 involves drawing from a deck of 52 cards, which includes 26 red cards (13 of which are face cards). The probability of drawing a red suited face-card on the first draw is 26/52 x 4/13 = 2/13, since there are 26 red cards out of 52 total cards, and 4 face-cards out of 13 cards of each suit. On the second draw, there will be one less card in the deck, so the probability of drawing another red suited face-card will be 25/51 x 3/12 = 5/204. The probability of drawing two red suited face-cards in a row is therefore (2/13) x (5/204) = 5/663.

Comparing the two events, we see that the probability of Event 1 is higher than that of Event 2. Therefore, rolling two 3's in a row using a six sided dice is the event with the higher theoretical probability.

(Please could you kindly mark my answer as brainliest you could also follow me so that you could easily reach out to me for any other questions)

(a) Option A, if its value is a large positive number, statistic A x2 provides strong evidence for the alternative hypothesis.

(b) Option B, the null hypothesis that gender is not related to personal goals and the alternative hypothesis that gender is related to personal goals.

(c) Option B, the variables considered in the chi-square test for the evaluation of contingency table B are categorical variables.

(a) If the x2 statistic is a large positive number, it provides strong evidence for the alternative hypothesis. The x2 statistic is used in hypothesis testing to determine if there is a significant difference between observed and expected frequencies. Large positive values ​​indicate that the observed frequency is significantly different from the expected frequency, supporting the alternative hypothesis.

(b) The hypotheses tested in the chi-square test for the contingency table are the null hypothesis that gender is not related to personal goals and the alternative hypothesis that gender is related to personal goals. personal goals. This test determines whether there is a significant relationship between two categorical variables.

(c) The variables considered in the chi-square test used to evaluate the contingency table are categorical variables. These variables cannot be assumed to be mean or normally distributed. The chi-square test is used to analyze the relationship between two or more categorical variables, where each variable has a discrete set of categories.

Complete Question:

For each problem, select the best response (a) A x2 statistic provides strong evidence in favor of the alternative hypothesis if its value is A. a large positive number. OB. exactly 1.96 c. a large negative number. D. close to o E. close to 1. (b) A study was performed to examine the personal goals of children in elementary school. A random sample of students was selected and the sample was given a questionnaire regarding achieving personal goals. They were asked what they would most like to do at school: make good grades, be good at sports, or be popular. Each student's sex (boy or girl) was also recorded. If a contingency table for the data is evaluated with a chi-squared test, what are the hypotheses being tested? A. The null hypothesis that boys are more likely than girls to desire good grades vs. the alternative that girls are more likely than boys to desire good grades. OB. The null hypothesis that sex and personal goals are not related vs. the alternative hypothesis that sex and personal goals are related. C. The null hypothesis that there is no relationship between personal goals and sex vs. the alternative hypothesis that there is a positive, linear relationship. OD. The null hypothesis that the mean personal goal is the same for boys and girls vs. the alternative hypothesis is that the means differ. O E. None of the above. (C) The variables considered in a chi-squared test used to evaluate a contingency table A. are normally distributed. B. are categorical. C. can be averaged. OD. have small standard deviations. E. have rounding errors.

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Looking at the back to back stem and leaf plot for the test scores, we can tell that D) The median and mean for Class A are higher than those for Class B. They have equal IQRs of 15.

The mean of Class A can be found to be :
= ( 70 + 70 + 75 + 75 + 80 + 80 + 85 + 85 + 85 + 85 + 90 + 90 + 90 + 95 + 95 + 95 + 95 + 95 + 95 + 100 + 100 + 100 ) / 22

= 92

The mean of Class B would be :

= ( 70 + 70 + 70 + 75 + 75 + 75 + 75 + 75 + 80+ 80 + 80 + 80 + 80 + 85 + 85 + 85 + 85 + 90 + 90+ 90 + 95 + 95 + 100 ) / 23

= 82

The mean of Class A is therefore larger than Class B. The IQR cannot be up to 70 given that the lowest value of 70 and the highest is 100 so the correct option is option D.

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The claim that the price of a suit decreased by 20% from £180 to £150 is incorrect. The actual decrease in price is about 16.7%.

The claim that the price of a suit decreased by 20% is incorrect.

A 20% decrease in price would mean that the new price is equal to 80% of the original price. So, if we take 80% of £180, we get:

0.8 x £180 = £144

Therefore, if the price of the suit decreased by 20%, the new price should be £144, not £150.

This means that the claim is incorrect and the price of the suit actually decreased by about 16.7% (i.e. (£180 - £150) / £180 x 100%).

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Answer: 12:30 P.M.

Step-by-step explanation:

50 + 50 + 50 + 90 minutes = 240 minutes
240 divided by 60 = 4
4 hours before 4:30 P.M. = 12:30 P.M.
Answer should be 12:30 P.M.

Answer:

They left the house at 1:20 P.M.

Step-by-step explanation:

The movie was an 1 1/2 so 4:30 - 1:30 = 3:00

Thy got there 50 minutes before so 3:00 - 0:50 = 2:10

It took them 50 minutes to get there so 2:10 - 0:50 = 1:20

Therefore they left the house at 1:20 P.M.

The statement that is NOT true is (C) The axis of symmetry is y = 4.

So, the rate of change of Logan's money remaining per ride paid for varies between -1/15 and -1/3, depending on the two rides that we choose to calculate the slope.

In mathematics, slope refers to the steepness of a line or a curve. More specifically, it is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line or curve.

The slope of a straight line can be found by dividing the change in the y-coordinates (vertical change) by the change in the x-coordinates (horizontal change) between any two points on the line. This is often represented by the symbol "m".

by the question.

To find the rate of change of Logan's money remaining per ride paid for, we need to calculate the slope of the line that represents the relationship between the number of rides paid for and the money remaining.

We can start by plotting the data points on a coordinate plane, where the x-axis represents the number of rides paid for and the y-axis represents the money remains.

(15, 10) (30, 9)

(20, 20) (50, 10)

We can see that the two points (15, 10) and (30, 9) represent the two rides where Logan had $10 and $9 remaining, respectively, and (20, 20) and (50, 10) represent the two rides where Logan had $20 and $10 remaining, respectively.

To calculate the slope of the line that passes through these points, we can use the formula:

[tex]slope = (y_2 - y_1) / (x_2 - x_1)[/tex]

where (x1, y1) and (x2, y2) are the coordinates of the two points.

Using the points (15, 10) and (30, 9), we get:

[tex]slope = (9 - 10) / (30 - 15) = -1/15[/tex]

Using the points (20, 20) and (50, 10), we get:

[tex]slope = (10 - 20) / (50 - 20) = -1/3[/tex]

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Answer:

Step-by-step explanation:

b is now located at (3,-12)

When Declan selects a pupil from the list at random 100 times, the estimation on the total number of times Declan selects a pupil who studies French is equal to 20 (times).

We have a table present above which shows the probability a pupil chosen at random studies Italian, French, Russian or German. Now, Declan randomly select a pupil from the list at 100 times. We have to estimate for the total number of times Declan will select a pupil that studies French. From the table, the probability to select a pupil who studies French = 0.2

So, Total number of times Declan will select a pupil that studies French

= 0.2 × 100

= 20 times

Hence, required estimation on total number of times is 20.

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Complete question:

The above table shows the probability a pupil chosen at random studies Italian, French, Russian or German.

Subject Italian French Russian German

Probability

Declan select a pupil from the list at random 100 times. Work out an estimate for the total number of times Declan will select a pupil that studies French

The equation of a line passing through the points (2,4) and (6,16) is

2) y=3x-2.

What is equation?

When the slope of the line being studied is known, and the provided point is also the y intercept, the slope intercept formula, y = mx + b, is utilised (0, b).   It provides both a slope and an intercept, which is why the form is known as the slope-intercept form.

Here the given two points are [tex](x_1,y_1)=(2,4)[/tex] and [tex](x_2,y_2)=(6,16)[/tex] .

Now using slope formula then,

=> slope m= [tex]\frac{y_2-y_1}{x_2-x_1}[/tex] = [tex]\frac{16-4}{6-2} =\frac{12}{4} = 3[/tex]

Now using equation formula then [tex]y-y_1=m(x-x_1)[/tex]

=> y-4=3(x-2)

=> y-4 = 3x-6

=> y = 3x-6+4

=> y = 3x-2

Hence the equation is 2) y = 3x-2.

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The answer of the given question is (a)  an additional 4 tiles are needed to surround the new perimeter. (b)  No, Percy's expression for the number of tiles is not equivalent to 4(s + 1).

An expression in mathematics is a combination of numbers, variables, and operations that represents a quantity or a value. Expressions can be simple or complex, and they can be used to represent anything from a single number to a complex algebraic equation. These expressions represent a sum of two numbers, a product of a number and a variable, a difference of a number and a constant, and the square root of a number, respectively.

a. Percy's table shows the number of tiles needed to surround a square pool with sides of length 1, 2, 3, 4, and 5 feet. He has calculated that for a pool with a side length of s feet, the number of tiles needed is given by the equation N = 8 + 4(s-1). From his table, he sees that as the side length of the pool increases by 1 foot, the number of border tiles needed increases by 4. This makes sense since each side of the pool gains 1 foot in length, so an additional 4 tiles are needed to surround the new perimeter.

b. No, Percy's expression for the number of tiles is not equivalent to 4(s + 1). Percy's equation is N = 8 + 4(s-1), which simplifies to N = 4s + 4. Stella's equation is 4(s + 1), which also simplifies to 4s + 4. Therefore,  two expressions are equivalent, and they both give number of tiles needed to surround  square pool with sides of length s feet.

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To produce the function f(x) = 2| x−1 | − 3 from the parent function f(x) = |x|, three transformations are required.

Horizontal shift: The function is shifted horizontally by 1 unit to the right. This is done by subtracting 1 from the input value of the function. The transformation is f(x) → f(x − 1).

Vertical stretch: The function is vertically stretched by a factor of 2. This is done by multiplying the output value of the function by 2. The transformation is f(x) → 2f(x − 1).

Vertical shift: The function is vertically shifted down by 3 units. This is done by subtracting 3 from the output value of the function. The transformation is f(x) → 2f(x − 1) − 3.

To find points on the function, we can substitute different values of x into the function and calculate the corresponding y-values. For example:

When x = 0: f(0) = 2|0 − 1| − 3 = -5

When x = 1: f(1) = 2|1 − 1| − 3 = -3

When x = 2: f(2) = 2|2 − 1| − 3 = -1

Using these points, we can plot the graph of the function f(x) = 2| x−1 | − 3. The graph is a V-shaped curve with the vertex at (1,-3), opening upwards. The points we found above can be plotted on the graph as (-1,-5), (1,-3), and (2,-1).

Answer:

Step-by-step explanation:

8x + 6y = 726

9x + 7y = 831

56x + 42y = 5082

-54x - 42y = -4986

2x = 95

x = 47.50 Shirt

47.50(8) + 6y = 726

380 + 6y = 726

6y = 346

y = 57.67 pants

In this case, x and y are both 27 degrees.

If I'm understanding you correctly, we have a semicircle with a triangle DEF engraved on it.

Vertex D has a 90 degree angle.

You are informed that angles D = 27 degrees, E = y degrees, and F = x degrees. You need to locate x and y.

Angle D is 90 degrees because triangle DEF is encircled by a semicircle.

Angles E and F therefore sum up to 90 degrees. Also, we are aware that angle D = 27° 2.

Angles E and F are so equal at 63 degrees and 27 degrees, respectively.

Half of a circle is a semicircle. It is a 2D shape that results from splitting a circle into two equal pieces. . A protractor, a circle folded in half, and other everyday objects are some instances of semicircles.

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The probability of renting exactly three apartments in a week using poisson distribution is 0.2107.

A Poisson distribution is a probability distribution that is used to evaluate how many times a particular event is likely to occur during a certain period of time.

The formula for calculating the probability of an event in a poisson distribution is as follows:

where e = 2.71828 (Euler's constant), x = the number of occurrences, μ = the expected value/average number of occurrences.In the given problem, the average number of new apartments rented per week is 2.3. Therefore,μ = 2.3Using the formula for Poisson distribution, we get:

P(3) = e^−2.3 * 2.3^3 / 3!

P(3) = 0.2107Therefore, the probability of renting exactly three apartments in a week is 0.2107.

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The p-value for this test is the probability of observing a z-score of -2.45 or lower, which is approximately 0.0072. Therefore, the answer is 0.0072.

The null hypothesis is a common arithmetic theory that asserts there is no statistical relationship and significance between two groups of observed data and measured phenomena for a particular set of single, observed variables.

What the researcher anticipates discovering after performing a study is expressed as a scientific hypothesis. A statistical hypothesis, on the other hand, is a prediction of what the statistician does NOT expect to discover. The term "NULL hypothesis" refers to this rationale.

We can use a one-sample proportion test with a one-tailed alternative to test this hypothesis, and calculate the p-value as the probability of observing 10 or fewer ones out of 120 rolls, assuming the null hypothesis is true.

Using a normal approximation to the binomial distribution.

(since n=120 is large and p0=1/6 is not too close to 0 or 1),

the test statistic is:

z = ([tex]\frac{10}{120}[/tex] - [tex]\frac{1}{6}[/tex]) / √([tex]\frac{\frac{1}{6} *\frac{5}{6} }{120}[/tex])

≈ -2.45

The p-value for this test is the probability of observing a z-score of -2.45 or lower, which is approximately 0.0072. Therefore, the answer is 0.0072.

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According to given information, the approximate p-value for this test is 0.0072.

What is meant by p-value?

In statistics, the p-value is the probability of observing a test statistic as extreme or more extreme than the one calculated from the sample, assuming that the null hypothesis is true.

To determine the p-value for this test, we need to perform a hypothesis test to determine if the observed proportion of ones in the 120 rolls is significantly different from the long-run proportion of ones. We can use the following hypotheses:

Null hypothesis: p = 1/6 (assuming a fair die)

Alternative hypothesis: p < 1/6 (the calculator is producing fewer ones than it should)

To perform the test, we can use the normal approximation to the binomial distribution, since np = 120(1/6) = 20 and n*(1-p) = 120*(5/6) = 100, which are both greater than 10.

The test statistic is:

z = (x - np) / sqrt(np(1-p))

where x = 10 is the number of ones observed, np = 20 is the expected number of ones under the null hypothesis, and p = 1/6 is the hypothesized proportion of ones.

Substituting these values, we get:

z = (10 - 20/6) / sqrt(20/6 * (1 - 1/6))

z = -2.305

The p-value for this test is the probability of observing a test statistic as extreme or more extreme than -2.305, assuming the null hypothesis is true. Since this is a one-tailed test in the left tail, the p-value is the area to the left of -2.305 under the standard normal distribution. Using a table or calculator, we can find that this area is approximately 0.0106.

Therefore, the approximate p-value for this test is 0.0106, which is closest to 0.0072.

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Answer:

[tex]6.25\pi \: {ft}^{2} [/tex]

Step-by-step explanation:

Given:

d = 5 ft

Find: a (area) - ?

First, we have to find the radius (r) of the circle, which is half diameter:

r = 0,5 × d

r = 0,5 × 5 = 2,5 ft

Now we can find the area of the circle (a):

[tex]a = \pi \times {r}^{2} = \pi \times ( {2.5})^{2} = 6.25\pi \: {ft}^{2} [/tex]

Answer:  The area of the circle is 19.625

Step-by-step explanation:

The formula for area of a circle is

[tex]A = \pi r^2[/tex]

We know that the diameter is 5 ft. The radius is half of the diameter, so we take 1/2 * 5 = 2.5

So the radius is 2.5

Now we will fill in the numbers for the formula

[tex]A = 3.14 * (2.5)^2[/tex]

or (3.14)(2.5)(2.5)

When we multiply all of the numbers, the area of the circle is 19.625

Using some trigonometric relations we can see that x  = 6.72

Here we know that the larger triangle is completely defined, then we can find the angle at X using a trigonometric relation:

Sin(a) = opposite cathetus/hypotenuse

Sin(a) = 7/25

Using the inverse sine function:

a = Asin(7/25) = 16.26°

Now, notice that angle at X is also an angle of the triangle whose x is a side, then we can use the relation:

sin(16.26°) = opposite cathetus/hypotenuse.

This time:

opposite cathetus = x

hypotenuse = 24

Replacing that:

sin(16.26°) = x/24

x = 24*sin(16.26°) = 6.72

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Answer:

y=-2x+6

I determined this by first calculating the slope which turned out to be -2, and then i solved using slope intercept formula y=mx+b. Slope is -2 because i input the 2 points 0,6 and 3,0 into slope formula

Step-by-step explanation:

Brainliest maybe ? :)

The complete factorization of 3x⁵- 75x³ is:- 3x³(x + 5)(x - 5)

How to solve factor?

To factor completely the expression 3x⁵ - 75x³, we can first factor out the greatest common factor (GCF) of the two terms, which is 3x³:

3x³(x² - 25)

Next, we can factor the expression inside the parentheses using the difference of squares formula:

3x³(x + 5)(x - 5)

Therefore, the complete factorization of 3x⁵ - 75x³ is:

3x³(x + 5)(x - 5)

We can check that this is the correct factorization by using the distributive property of multiplication and verifying that the product of the factors is equal to the original expression:

3x³(x + 5)(x - 5) = 3x³(x² - 25)

= 3x³x² - 3x³(25)

= 3x⁵ - 75x³

So the factorization is correct.

In summary, to factor completely an expression like 3x⁵ - 75x³, we should first factor out the GCF and then look for further factorization opportunities using various factorization techniques such as the difference of squares formula, the sum or difference of cubes formula, or the quadratic formula. It's important to remember to check our work by multiplying the factors back together to ensure that we get the original expression.

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The value of AD of the triangle using trigonometric ratios is: 18.7

The sum of angles in a triangle is 180 degrees. Thus:

∠ABC = 180 - (41 + 35)

∠ABC = 104°

Using sine rule, we have:

AB/sin 35 = AC/sin 104

AB = (42 * sin 35)/(sin 104)

AB = 42 * 0.5911

AB = 24.826

Now, using trigonometric ratio:

AD/AB = cos 41

AD = 24.826 * 0.7547

AD = 18.7

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Answer:

27 cm^2

Step-by-step explanation:

Area of a triangle is B * H / 2

6 * 9 / 2 = 27