If a sample of 50 items are chosen at random. the probability that the sample mean length is greater than 9.1 inches is 0.0571, or 5.71%.
How to find the probability?The sample mean of the 50 items follows a normal distribution with mean equal to the population mean (μ), and standard deviation equal to σ/√n, where n is the sample size.
Substituting the given values, we have:
= μ = 9 inches
= σ/√n = 1/√50 inches
Now, we need to standardize the sample mean distribution to find the corresponding z-score using the formula:
z = (X- μX) / σX
Substituting the given values, we have:
z = (9.1 - 9) / (1/√50) = 1.58
The probability of getting a z-score of 1.58 or less is 0.9429. Therefore, the probability of getting a z-score of 1.58 or greater is:
P(z > 1.58) = 1 - P(z ≤ 1.58) = 1 - 0.9429 = 0.051
Therefore the probability that the sample mean length is greater than 9.1 inches is 0.0571.
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find the missing value to the nearest hundredth cos__=7/18
67.11
37.67
22.89
21.25
The missing value is approximately 64.12 degrees, which is closest to option D: 21.25 using inverse cos.
The inverse cosine (or arccosine) of both sides of the equation must be taken in order to locate the missing value in the equation cos = 7/18 to the closest tenth. We will then be able to convert the value of to degrees and round it to the closest hundredth using the value of the radian that is produced.
cos θ = 7/18
= arccos(7, 18, 7)
A 1.1181 radian angle
We can use the conversion factor /180 to convert radians to degrees.
θ = 1.1181 × 180/π
64.12 degrees.
As a result, the figure that is lacking is roughly 64.12 degrees, which is closest to choice D: 21.25.
Due to the periodic nature of the cosine function, there are often two solutions when determining the inverse cosine of a value. But, since the problem states that we should round to the closest hundredth, which suggests a single answer, we can disregard the second option in this instance. By reinserting the value we discovered into the original equation and making sure it equals 7/18, we can further confirm that the result is accurate.
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FIND THE MISSING SIDES OF THE KITE
Picture for more info!
Thank you.
The required sides of kite are √61, √61, √221 and √221 units.
What is Pythagoras theorem?According to the Pythagoras theorem, the square of the hypotenuse in a right-angled triangle equals the total of the squares of the other two sides. The formula for this theorem is c² = a² + b², where c is the hypotenuse and a and b are the triangle's two sides. Pythagoras theory triangles are another name for these triangles.
According to question:In ΔAOD
AD = [tex]\sqrt{5^2+6^2} = \sqrt{61}[/tex] units
In ΔAOB
AB = [tex]\sqrt{5^2+6^2} = \sqrt{61}[/tex] units
In ΔBOC
BC = [tex]\sqrt{5^2+14^2} = \sqrt{221}[/tex] units
In ΔDOC
DC = [tex]\sqrt{5^2+14^2} = \sqrt{221}[/tex] units
Thus, required sides of kite are √61, √61, √221 and √221 units.
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if a college instructor alters the distribution of his or her students' midterm exam grades because the class did not do as well as last year's class, with what type of standards is the instructor most concerned?
The college instructor is most concerned with equity standards.
Equity standards are when an instructor adjusts grades to ensure fairness to all students regardless of their individual performances.
This means that if a class as a whole does not do as well as last year, the instructor will alter the distribution of grades so that the overall grades reflect the effort put in by the class.
For example, if the class average is a C, the instructor may raise some grades to a B or even A in order to ensure that the grades are fair to all students in the class.
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1. The orthocenter is outside the triangle. 2. An altitude is the same line segment as an angle bisector. 3. If the perpendicular bisector of one side of a triangle intersects the opposite vertex, then the triangle is isosceles
The solution of the match is
(1) Orthocenter (C) The point at which the altitudes of a triangle
meet.
(2) Centroid (D) The point at which the medians of a triangle
meet.
(3) Circumcenter (A) The point of intersection of the perpendicular
bisectors of the sides of a triangle.
(4) Incentre (B) The point at which the three angle bisectors
of a triangle meet.
The orthocenter of a triangle is the point where the altitudes of the triangle intersect. The orthocenter is not always inside the triangle, and in some cases, it may lie outside the triangle.
The centroid of a triangle is the point where the three medians of the triangle intersect. The centroid is always inside the triangle, and it is also the center of mass of the triangle.
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect. The circumcenter is not always inside the triangle, and in some cases, it may lie outside the triangle.
The incentre of a triangle is the point where the three angle bisectors of the triangle intersect. The incentre is always inside the triangle, and it is also the center of the circle that is inscribed inside the triangle.
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Complete Question:
Match the following.
(1) Orthocenter (A) The point of intersection of the perpendicular
bisectors of the sides of a triangle.
(2) Centroid (B) The point at which the three angle bisectors of
a triangle meet.
(3) Circumcenter (C) The point at which the altitudes of a triangle
meet.
(4) Incentre (D) The point at which the medians of a triangle
meet.
what is the probability of rolling a sum less than or equal to 7 ? express your answer as a fraction or a decimal number rounded to four decimal places.
The probability of rolling a sum of 7 or less on two dice is 11/36. This can be expressed as a decimal number rounded to four decimal places as 0.3056.
We need to look at the possibilities when rolling two dice. Each die has six sides, which means the total number of outcomes when rolling two dice is 36. There are 11 combinations of two dice that produce a sum of 7 or less: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (3,1), (3,2). So, 11/36 can be simplified to 1/4, or 0.3056.
To understand this in a more visual way, you can draw a table and use colored dots to indicate the possible combinations. This will show you that out of 36 total possibilities, 11 of them produce a sum of 7 or less.
In summary, the probability of rolling a sum of 7 or less on two dice is 11/36, which can be expressed as a decimal number rounded to four decimal places as 0.3056.
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12. STEM An engineer is designing solar
panels for a rover that will explore
Mars. The solar panel unfolds to an
isosceles trapezoid. The length of
one leg is 0.1.x -0.5 meters, and the
length of the other leg is 0.3.x-2.1
meters. What is the length of each of
the legs of the solar panel?
Answer:
Step-by-step explanation:
Let's call the length of the shorter leg of the isosceles trapezoid "a" and the length of the longer leg "b".
From the problem, we know that:
a = 0.1x - 0.5 (length of one leg is 0.1.x - 0.5 meters)
b = 0.3x - 2.1 (length of the other leg is 0.3.x-2.1 meters)
Since we know that the trapezoid is isosceles, we know that a = b. So we can set the two expressions equal to each other:
0.1x - 0.5 = 0.3x - 2.1
Now we can solve for x:
0.2x = 1.6
x = 8
Now that we know x, we can substitute it back into the expressions for a and b to find their lengths:
a = 0.1(8) - 0.5 = 0.3 meters
b = 0.3(8) - 2.1 = 0.3 meters
Therefore, each of the legs of the solar panel are 0.3 meters long.
5. Janice is creating a design that is part of a logo consisting of a small and a large circle.
The diameter of the small circle is of the diameter of the large circle. The diameter of
the large circle is 27 centimeters. To the nearest tenth of a centimeter, what is the area
of the smaller circle?
27 cm
Answer: The area of the smaller circle is [tex]65.6cm^{2}[/tex].
Step-by-step explanation:
A = [tex]\pi r^{2}[/tex], r is the radius.
1. Find out the diameter of the small circle.
[tex]\frac{1}{3}(27)= 9[/tex]
2) Find out the radius, r.
[tex]radius = \frac{diameter}{2} =\frac{9}{2} =4.5[/tex]
3) We plugin r = 4.5 into the formula to solve the area of the smaller circle.
[tex]A=\pi r^{2}=\pi (4.5)^{2} =65.6cm^{2}[/tex]
HELP QUICK PLEASE! DUE TONIGHT!
Josh asked you to help him understand interpolation and extrapolation.
Use an example and a graph to help explain how interpolation and
extrapolation are similar and how they are different
Answer:
Interpolation and extrapolation are two methods used to estimate data points within or beyond a given set of values.Interpolation is the process of estimating a data point within the given range of values, based on the relationship between the known data points. For example, suppose we have the following data points: (1, 3), (2, 5), and (4, 9). If we want to estimate the value of y for x = 3, we can use interpolation to calculate it based on the trend of the data points within the given range. In this case, we can see that the slope of the line between (2, 5) and (4, 9) is the same as the slope of the line between (1, 3) and (2, 5). Therefore, we can estimate the value of y for x = 3 to be 7, using the trend of the known data points.Extrapolation, on the other hand, is the process of estimating a data point beyond the given range of values, based on the trend of the known data points. For example, suppose we have the same data points as before: (1, 3), (2, 5), and (4, 9). If we want to estimate the value of y for x = 5, we can use extrapolation to calculate it based on the trend of the known data points. In this case, we can see that the slope of the line between (2, 5) and (4, 9) is the same as the slope of the line between (1, 3) and (2, 5). Therefore, we can estimate the value of y for x = 5 to be 11, assuming that the trend of the known data points continues beyond the given range.Here is a graph that shows both interpolation and extrapolation:
{graph attached below}
In the graph, the blue dots represent the known data points. The red line represents the trend of the known data points, which can be used for interpolation and extrapolation. The green dot represents an interpolated data point, while the purple dot represents an extrapolated data point.In summary, interpolation and extrapolation are similar in that they both involve estimating data points based on the trend of the known data points. However, they differ in that interpolation estimates data points within the given range of values, while extrapolation estimates data points beyond the given range of values.
hope this helps!
Each voter from a random sample of 334 registered voters was asked their impression of two candidates running for the same national office. The
table summarizes the responses.
Which of the following is the me
Candidate A?
Favorable
Unfavorable
No opinion
Have not heard of
Candidate A. Candidate B
Favorable 138 200
Unfavorable 44 47 No Opinion 88 47 Haven’t heard of 64 40
Which of the following is the most appropriate method to use to estimate the proportion of all registered voters who have a favorable impression of Candidate A?
A. A one-sample z-interval for estimating a sample proportion
B. A one-sample z-interval for estimating a population proportion
C. A matched-pairs t-interval for estimating a mean difference
D. A two-sample z-interval for estimating a difference between sample proportions
E. A two-sample z-interval for estimating a difference between population proportions
The answer choice that is the most appropriate is B. A one-sample z-interval for estimating a population proportion
Why is this the most appropriate?
This is the most appropriate method because you are trying to estimate the proportion of all registered voters who have a favorable impression of Candidate A based on the sample data.
A one-sample z-interval is used when you want to estimate a population proportion using a sample proportion.
To calculate the one-sample z-interval for estimating a population proportion, you can use the following formula:
Confidence Interval = p ± Z * sqrt((p * (1 - p)) / n)
where:
p is the sample proportion (favorable opinions of Candidate A / total voters in the sample)Z is the critical value corresponding to the desired level of confidence (e.g., 1.96 for a 95% confidence interval)n is the sample size (334 in this case)sqrt represents the square root functionUsing this method, you will calculate a confidence interval that estimates the true proportion of registered voters who have a favorable impression of Candidate A, based on the sample data.
This is the most appropriate method because it focuses on estimating a single population proportion and incorporates the sample data and sample size into the calculation.
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a study was conducted to evaluate the stress level of senior business students at a particular college. forty students were selected at random from the senior business class, and their stress level was monitored by attaching an electrode to the frontalis muscle (forehead). for the forty students, the mean emg (electromyogram) activity was found to be 35.8 microvolts. in addition, the standard deviation of the emg readings was found to be 2.5 microvolts. what would be the 99% confidence interval on the true mean emg activity for all seniors in the class? (you should be using statistical software such as statcrunch.) group of answer choices [34.7296, 36.8704] [34.7296, 36.7804] [34.7456, 36.0566] [34.9672, 36.7840]
The 99% confidence interval on the true mean EMG activity for all seniors in the class is option (a) [34.7296, 36.8704]
To calculate the 99% confidence interval for the true mean EMG activity for all seniors in the class, we can use the formula
CI = X ± Zα/2 (σ/√n)
where
X = sample mean = 35.8
Zα/2 = the critical value for the 99% confidence interval, which can be found using a standard normal distribution table or calculator. For a two-tailed test, α/2 = 0.005, so Zα/2 = 2.576.
σ = sample standard deviation = 2.5
n = sample size = 40
Substituting these values into the formula, we get
CI = 35.8 ± 2.576 (2.5/√40) = [34.7296, 36.8704].
Therefore, the correct option is (a) [34.7296, 36.8704]
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Given that New Mexico has a population of about 12 people per square miles and
an area of about 120,000 square miles, what is the population of New Mexico
Mark only one oval.
10,000
1,000,000
1,440,000
2,400,000
Answer:
1440000
Step-by-step explanation:
Since for every mile, there are 12 people, we have 120000 miles. If we multiply 12 by 120000, we get 1440000.
What is the MAD of 12, 10, 12, 6, 8, 4, 2, 12,
The mean absolute deviation (MAD) of the given set of data is 3.875.
To calculate the mean absolute deviation, we first need to find the mean of the data set by adding up all the numbers and dividing by the total number of values:
Mean = (12 + 10 + 12 + 6 + 8 + 4 + 2 + 12) / 8
Mean = 8
Next, we find the absolute deviation of each number from the mean. To do this, we subtract the mean from each number and take the absolute value:
|12 - 8| = 4
|10 - 8| = 2
|12 - 8| = 4
|6 - 8| = 2
|8 - 8| = 0
|4 - 8| = 4
|2 - 8| = 6
|12 - 8| = 4
Then, we find the mean of the absolute deviations:
Mean absolute deviation = (4 + 2 + 4 + 2 + 0 + 4 + 6 + 4) / 8
Mean absolute deviation = 26 / 8
Mean absolute deviation = 3.875
Therefore, the mean absolute deviation of the given set of data is 3.875.
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Complete Question:
what is the mean absolute deviation of the following set of data 12 10 12 6 8 4 2 12.
. in a classroom of 30 students, 3 of the students wear wrist watches. (a) if 14 students are selected with replacement, what is the probability that exactly 2 of them wear wrist watches? (b) if 14 students are selected without replacement,
Probability of
=0.0403.
The probability that exactly two of the 14 students selected with replacement will wear wrist watches is 3/30 * 2/29 * 27/28 * 26/27 = 0.0437.
The probability that exactly two of the 14 students selected without replacement will wear wrist watches is 3/30 * 2/29 * 26/28 * 25/27 = 0.0403.
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The size of a computer screen is determined by the length of its diagonal. Each student in Mrs. W's math class
was asked to measure the diagonal of his or her Chromebook. Jason's measurements was 13 inches. The
computer's actual diagonal length is 14 inches. Calculate the percent error for Jason's measurement.
The percent error for Jason's measurement is 7.14%. This means that his measurement was off by 7.14% compared to the actual value.
What is percent error?Percent error is a measure of the difference between an actual value and an estimated or measured value, expressed as a percentage of the actual value. It is used to evaluate the accuracy of measurements or calculations.
According to question:To calculate the percent error for Jason's measurement, we first need to find the absolute error, which is the difference between his measurement and the actual value:
Actual value minus measured value equals absolute error.
Absolute error = |14 - 13|
Absolute error = 1 inch
Next, we can use the formula for percent error:
(Absolute error / Actual Value) x 100% equals the percent of error.
Percent error = (1 / 14) x 100%
Percent error = 0.0714 x 100%
Percent error = 7.14%
Therefore, the percent error for Jason's measurement is 7.14%. This means that his measurement was off by 7.14% compared to the actual value.
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d. if a student was an undergraduate business major, what is the probability that the student intends to attend classes full-time in pursuit of an mba degree (to decimals)?
0.4
The probability that an undergraduate business major intends to attend classes full-time in pursuit of an MBA degree depends on the individual student's preferences and situation. Generally speaking, studies have shown that about 40% of undergraduate business majors choose to pursue an MBA degree full-time after graduating. Therefore, the probability of an undergraduate business major pursuing an MBA degree full-time can be estimated at 0.4.
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Rewrite the expression 2(10+12) using the distributive property of multiplication over addition
The expression 2(10+12) can be rewritten as 44 using the distributive property of multiplication over addition.
The distributive property is a fundamental property of arithmetic that relates multiplication to addition and subtraction. The distributive property of multiplication over addition states that for any numbers a, b, and c
a( b + c ) = ab + ac
Using this property, we can rewrite the expression 2(10+12) as
2( 10 + 12 ) = 2(10) + 2(12)
Here, we have distributed the factor of 2 over the terms inside the parentheses, using the distributive property. This gives us:
2(10+12) = 20 + 24
Simplifying further, we get
2(10+12) = 44
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A sports scout for a team is looking for the more consistent player. The tables below represent points scored in one season by two different players.
Player A
3 rows
and 5 columns
10, 12, 6, 10, 13, 8, 12, 3, 21, 14, 7, 0, 15, 6, 16
Player B
3 rows and 5 columns
10, 3, 12, 26, 20, 24, 25, 26, 5, 48, 24, 18, 20, 27, 25
Compare the data in the tables. Which player should the scout choose? Explain your answer.
Based on the given information, the scout should choose Player A if they are looking for consistency.
What are range and standard deviation?
Range is the difference between the highest and lowest scores in the set.
Standard deviation measures how spread out the scores are from the mean.
To determine which player is more consistent, we need to look at the variability in their scores. One way to measure variability is to calculate the range and the standard deviation of each player's scores.
The smaller the range, the more consistent the player's scores are.
A smaller standard deviation indicates that the scores are closer together, which means the player is more consistent.
Using the data provided, we can calculate the range and standard deviation for both players:
Player A:
Range = 21 - 0 = 21
Standard deviation = 5.70
Player B:
Range = 48 - 3 = 45
Standard deviation = 10.89
From these calculations, we can see that Player A has a smaller range and a smaller standard deviation, which means that their scores are more consistent than Player B's. Therefore, the scout should choose Player A if they are looking for consistency.
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write down a model that allows the variance of u to differ between men and women. the variance should not depend on other factors.
The model will be as follows: y = b0 + b1x1 + b2x2 + u
To write a model that allows the variance of u to differ between men and women, the best option is to use the heteroscedasticity model. This model will enable the variances to differ between the genders without depending on other factors.
The model will be as follows: y = b0 + b1x1 + b2x2 + u, Where y = dependent variable, b0 = constant or intercept, b1 and b2 are the coefficients for the independent variables x1 and x2, u = error term for the model. However, this model does not allow the variance of u to differ between men and women.
To achieve this, we need to modify the model to allow the variance of u to differ between men and women. This can be done by using the weighted regression model. The model will be as follows: y = b0 + b1x1 + b2x2 + u, where y = dependent variable, b0 = constant or intercept, b1 and b2 are the coefficients for the independent variables x1 and x2, u = error term for the model.
The weights can be represented as: wi = 1/σ^2i, where: wi = weight for observation, iσi = standard deviation of the ith observation. This model will allow the variance of u to differ between men and women without depending on other factors.
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Can yall help me out?
the answer is A: 63 square units
Answer:
A (Im not entirely sure if this is correct)
Step-by-step explanation:
how I got this answer is: I found the area of the two triangles which was 12.5 (because the area of a triangle is 1/2(b*h) and the base is 3 and the height is 7 then you divide that by two) and I added them up to get 21. Then you would multiply 3 by two because a triangle is half of a rectangle and you would do b*h for a rectangle area and then multiply that by 7 ot get 42 and you would add taht to 21 to get 63
given the arithmetic sequence what is the domain for N?
The domain for n in the arithmetic sequence is the one in the first option:
all integers where n ≥1
What is the domain of N?We have a arithmetic sequence defined by the general formula:
aₙ = 4 - 3*(n - 1)
And we want to find the domain for the possible values of n that we can use in that formula.
Remember that the terms of a sequence are defined as:
a₁, a₂, a₃,...
So the values of n are positive whole numbers, in this case the first one is n = 1.
a₁ = 4 + 3*(1 - 1)
a₁ = 4
And then we can keep using any positive integer, then the correct option is:
all integers where n ≥1
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6x6x6x6x6x6x6 devided by 6x6x6x6x6x6 in indext form
The result of 6x6x6x6x6x6x6 divided by 6x6x6x6x6x6 in index form is 6.
In index form, 6x6x6x6x6x6x6 divided by 6x6x6x6x6x6 can be written as:
(6⁷) / (6⁶).To divide two exponential expressions with the same base (in this case, 6), we can subtract the exponents:
6⁽⁷⁻⁶⁾ = 6¹ = 6.
Index form, also known as exponential form, is a way of writing numbers or expressions using exponents. In index form, a number is expressed as a base raised to a power or exponent.
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What is the product of (3x+2) and (x - 7) ?
The diameter of a softball is 3.8 inches. Estimate the volume within the softball. Round answers to the nearest
whole number and leave in the answer.
a. Volume: 73
b. Volume: 14
c. Volume: 9
d. Volume: 58
The volume of the ball is 9π (optionC)
What is volume of a sphere?A sphere is symmetrical, round in shape. It is a three dimensional solid, that has all its surface points at equal distances from the center.
The ball takes the shape of the sphere. The volume of a sphere is therefore calculated as ;
V = 4/3 πr³
r = d/2 = 3.8/2 = 1.9
V = 4/3 × π × 1.9³
V = 27.44 π/3
V = 9π ( nearest whole number)
therefore the volume of the ball is 9π (nearest whole number)
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w(x)= -5(x-8)(x+4)
What is the maximum height that the stone will reach?
Answer:
(2, 180)
Step-by-step explanation:
w(x)=-5x^2+20x+160
a=-5 b=20
x=-20/2x(-5)
x=2
w(2)=180
2, 180
when researching a population where it is impossible or impractical to compile a list of the elements, which sampling technique is appropriate to use?
When researching a population where it is impossible or impractical to compile a list of the elements, the sampling technique that is appropriate to use is known as the cluster sampling technique
Cluster sampling is a type of probability sampling that is frequently used when the population is too large to be measured as a whole. The population is divided into smaller, more manageable clusters that are less difficult to study than the entire population.Therefore, cluster sampling is an excellent alternative when the target population cannot be measured because it is too large, too vast, or too hard to reach. In this technique, clusters are chosen by a random process or some other procedure.The final sampling method is achieved by selecting samples from within the clusters, which are frequently chosen using simple random sampling or some other appropriate method. When it comes to cluster sampling, it's important to remember that the clusters chosen must be heterogeneous internally but homogeneous externally.
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in a standard deck of 52 cards (4 suits, 13 numbers per suit), how many hands can we be dealt 5 cards that do not share a number
Answer: 1287
Step-by-step explanation:
Near Pittsburgh, a cable car transports people from the Monongahela River valley to the community on top of the overlooking bluff. The Monongahela Incline has a grade of 78%. Find the measure of the angle the cable car track makes. with the valley floor. If necessary, round to the nearest degree.
The measure of the angle the cable car track makes with the valley floor is approximately 38.7 degrees.
What is grade?The slope or inclination of a line or surface is sometimes referred to as the grade in mathematics. The ratio of the vertical change (rise) to the horizontal change (run) between any two locations on a line or surface is what is meant by this term. Usually, the grade is given as a percentage or a fraction. In engineering, building, and transportation, grades are frequently used to define the slope of roads, railroads, and other infrastructure. The grade, which refers to how steep the slope is when referring to a cable car, is a crucial element in determining the safety and effectiveness of the system.
To find the angle we use the trigonometric functions:
tan(θ) = grade = rise/run
Given that, the grade of the incline is 78%, then grade = 0.78.
θ = arctan(0.78) ≈ 38.7 degrees.
Hence, the measure of the angle the cable car track makes with the valley floor is approximately 38.7 degrees.
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Determine the equation of the circle graphed below.
Answer:
Step-by-step explanation:
Circle centre: (4,7)
Radius: 3
Equation is:
[tex](x-h)^2+(y-k)^2=r^2[/tex] (where (h,k) is center, r is radius)
[tex](x-4)^2+(y-7)^2=3^2[/tex]
[tex](x-4)^2+(y-7)^2=9[/tex] (This is the general standard form )
in expanded form,
[tex](x^2-8x+16)+(y^2-14y+49)=9[/tex]
[tex]x^2+y^2-8x-14y+56=0[/tex]
Note: Unless stated either form is an acceptable answer.
A plant in my garden is growing 2/3 of a foot every 3/4 of a month. How much does it grow per month?
The plant is growing 0.89 feet per month.
To calculate how much a plant grows per month, you need to first convert the measurements into one unit of measurement.
Since the plant is growing 2/3 of a foot every 3/4 of a month,
we can first convert the fractions of a foot and a month into decimal equivalents.
2/3 of a foot is 0.67 feet, and 3/4 of a month is 0.75 months.
We can then calculate the amount of growth per month by dividing the amount of growth per 3/4 of a month by the amount of time that is 3/4 of a month (0.75).
The equation is 0.67 feet / 0.75 months = 0.89 feet per month.
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Please help!
Question below!!
The coordinates for the dilated triangle is obtained as A'(-1, 8), B'(-7, 6), and C'(-7, -4).
What is dilation?
Resizing an item uses a transition called dilation. Dilation is used to enlarge or contract the items. The result of this transformation is an image with the same shape as the original. However, there is a variation in the shape's size. The initial shape should be stretched or contracted during a dilatation.
The coordinates point of the triangle is given as -
A (-1,6)
B (-4,5)
C (-4,0)
To dilate triangle ABC by a scale factor of 2 about the point (-1, 4), we can follow these steps -
Translate the triangle so that the center of dilation is at the origin.
We can do this by subtracting (-1, 4) from each vertex -
A' = (-1, 6) - (-1, 4) = (0, 2)
B' = (-4, 5) - (-1, 4) = (-3, 1)
C' = (-4, 0) - (-1, 4) = (-3, -4)
Dilate the translated triangle by multiplying the coordinates of each vertex by the scale factor of 2 -
A'' = 2(0, 2) = (0, 4)
B'' = 2(-3, 1) = (-6, 2)
C'' = 2(-3, -4) = (-6, -8)
Translate the dilated triangle back to its original position by adding (-1, 4) to each vertex -
A'B'C' = A'' + (-1, 4) = (-1, 8)
B'' + (-1, 4) = (-7, 6)
C'' + (-1, 4) = (-7, -4)
Therefore, the coordinates of the dilated triangle A'B'C' are (-1, 8), (-7, 6), and (-7, -4).
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