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The two triangles are related by SAS criteria, so the triangles are congruent.

Congruent triangles are triangles that are precisely the same size and form. When the three sides and three angles of one triangle match the same dimensions as the three sides and three angles of another triangle, two triangles are said to be congruent. Corresponding portions are those areas of the two triangles that share the same dimensions (are congruent). This indicates that corresponding triangle parts are congruent (CPCTC).

From the given figure we observe for that the two triangles two sides and the corresponding angle of 90 degree is similar.

Thus, using the SAS criteria we see that the two triangles are equal.

Hence, the two triangles are related by SAS criteria, so the triangles are congruent.

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This group is isomorphic to the group of invertible matrices GL(n) if the vector space has finite dimension n.

Matrices are rectangular arrays of numbers or other mathematical objects, such as complex numbers, polynomials, or functions. Matrices are an important tool in many branches of mathematics, including linear algebra, calculus, and statistics. A matrix is usually written with square brackets enclosing its elements, with rows and columns separated by commas or spaces.

by the question.

Yes, you are correct! The set of all n x n matrices with entries in a field (such as the real numbers) is a group under addition, known as the general linear group GL(n), where the operation is matrix addition. This group satisfies the four group axioms:

Closure: The sum of two matrices is also a matrix.

Associativity: Matrix addition is associative.

Identity: The zero matrix is the identity element of GL(n).

Inverse: For every matrix A in GL(n), there exists a matrix -A such that A + (-A) = 0.

However, the set of all n x n matrices with entries in a field is not a group under multiplication because matrix multiplication is not always commutative and not every matrix has an inverse with respect to matrix multiplication.

On the other hand, the set of invertible n x n matrices, denoted by GL(n) or GL (n, F) when the field is specified, does form a group under matrix multiplication. The group axioms are satisfied:

Closure: The product of two invertible matrices is invertible.

Associativity: Matrix multiplication is associative.

Identity: The identity matrix I_n is the identity element of GL(n).

Inverse: For every invertible matrix A in GL(n), there exists an inverse matrix A^-1 such that A * A^-1 = I_n.

Similarly, the set of invertible linear transformations of a vector space into itself forms a group under composition of functions, known as the general linear group GL(V) of the vector space V.

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Answer:5

Step-by-step explanation:5555555

Answer:

y = x + 2

Step-by-step explanation:

The slope intercept form is y = mx + b

m = the slope

b = y-intercept

Slope = rise/run or (y2 - y1) / (x2 - x1)

Pick 2 points (0,2) (-2, 0)

We  see the y decrease by 2, and the x decrease by 2, so the slope is

m = -2 / -2 = 1

Y-intercept is located at (0,2)

So, the equation is y = x + 2

Answer:

x = ±10

Step-by-step explanation:

1) Subtract 10 from both sides.

[tex]-0.1 \times x^2=-10[/tex]

2) Divide both sides by -0.1.

[tex]x^2=\frac{-10}{-0.1}[/tex]

3) Simplify [tex]\frac{-10}{-0.1}[/tex] to 100.

[tex]x^2=100[/tex]

4) Take the square root of both sides.

[tex]x=\pm \sqrt{100}[/tex]

5) Since 10 * 10 is 100, the square root of 100 is 10.

[tex]x=\pm10[/tex]

Answer:

[tex] {y}^{6} {x}^{ - 2} [/tex]

Step-by-step explanation:

4 will get multiplied to the exponents of x and y inside the bracket.

The 99% confidence interval for the population proportion p is :lower bound= 0.225, upper bound 0.375

What does a confidence interval actually mean?

Your estimate's mean is added to and subtracted from by the estimate's range to create a confidence interval. If you repeat your test, within a specific level of confidence, this is the range of values you anticipate your estimate to fall within.

Given that,

n = 250

x = 75

Point estimate = sample proportion =  p = x / n = 75/250=0.3

1 -  p =1 - 0.3=0.7

At 99% confidence level the z is ,

α  = 1 - 99% = 1 - 0.99 = 0.0

α / 2 = 0.01 / 2 = 0.005

Z /2 = Z0.005 = 2.576  (  Using z table   )  

Margin of error = E = Zα / 2 * √(( p * (1 -  p)) / n)

                              = 2.576 (√((0.3*0.7) /250 )

                            E  = 0.075

A 99% confidence interval for population proportion p is ,

p - E < p <  p + E

0.3 -0.075 < p < 0.3 + 0.075

0.225< p < 0.375

The 99% confidence interval for the population proportion p is :lower bound= 0.225, upper bound 0.375.

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Answer:

[tex]14 {y}^{2} \sqrt{ {x}^{3} {z}^{9} }[/tex]

Step-by-step explanation:

[tex] \sqrt{196 {x}^{3} {y}^{4}{z}^{9} }= \sqrt{196} \times \sqrt{ {x}^{3} } \times \sqrt{ {y}^{4} } \times \sqrt{ {z}^{9} } \\ \sqrt{196} = 14 \\ \sqrt{ {x}^{3} } = {x}^{ \frac{3}{2} } \\ \sqrt{ {y}^{4} } = {y}^{2} \\ \sqrt{ {z}^{9}} = {z}^{ \frac{9}{2} }[/tex]

A fractional exponent is not necessarily simpler so just take out the 1st and 3rd parts of the term which simplify nicely:

[tex] \sqrt{196 {x}^{3} {y}^{4}{z}^{9} } = 14 {y}^{2} \sqrt{ {x}^{3} {z}^{9} } [/tex]

In response to the stated question, we may state that The equivalent trigonometry expression of 7/tan b + 7tan b is (7 - 7tan b cot b)/sin b.

The study of the connection between triangle side lengths and angles is known as trigonometry. The concept first originated in the Hellenistic era, during the third century BC, due to the application of geometry in astronomical investigations. The subject of mathematics known as exact techniques deals with certain trigonometric functions and their possible applications in calculations. There are six commonly used trigonometric functions in trigonometry. Sine, cosine, tangent, cotangent, secant, and cosecant are their separate names and acronyms (csc). The study of triangle characteristics, particularly those of right triangles, is known as trigonometry. As a result, geometry is the study of the properties of all geometric forms.

tan(A + B) = (tan A + tan B)/(1 - tan A tan B)

Set A = 90 degrees and B = b degrees:

tan(90 + b) = (tan 90 + tan b)/(1 - tan 90 tan b)

tan(90 + b) = (undefined + tan b)/(1 - undefined tan b)

tan(90 + b) = -cot b

7/tan b + 7tan b

= 7/(tan b) + 7(tan(90 + b) - 1)

= 7/(tan b) + 7(-cot b - 1)

= (7 - 7tan b cot b)/sin b

The equivalent expression of 7/tan b + 7tan b is (7 - 7tan b cot b)/sin b.

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Answer:

m = 6

Step-by-step explanation:

Slope = rise/run or (y2 - y1) / (x2 - x1)

Pick 2 points on the table (4,6) (5,12)

We see the y increase by 6 and the x increase by 1, so the slope is

m = 6

So, the slope is 6

The distance  from Link to the Octorok  is 10.63 units.

We know that the distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula below:

distance = √( (x₂ - x₁)² + (y₂ - y₁)²)

Here we want to find the distance from Link to the Octorok so Link can attack, so we need to get the distance between the points (-4, -5) and (3, 3).

The distance will be:

distance = √( (3 + 4)² + (3 + 5)²)

distance = √( (7)² + (8)²)

distance = √113

distance = 10.63

The distance is 10.63 units.

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Answer: The probability of drawing a vowel from a deck of 12 cards is 5/12, since there are 5 vowels (A, E, I, O, U) and 12 cards in total.

Since Brandi returns the first card drawn to the deck before drawing the second card, the outcome of the first draw does not affect the outcome of the second draw. Therefore, we can treat the two draws as independent events.

The probability of drawing a vowel on the first draw is 5/12, and the probability of drawing a vowel on the second draw is also 5/12. Since we want to find the probability of both events happening (drawing a vowel both times), we can multiply the probabilities:

P(vowel on both draws) = P(vowel on first draw) * P(vowel on second draw)

= (5/12) * (5/12)

= 25/144

Therefore, the theoretical probability that Brandi draws a card with a vowel both times is 25/144, which is approximately 0.1736 or 17.36%.

So the closest answer choice is B) 2.08%.

Step-by-step explanation:

The number of 5 fluid ounce portions needed in 20 gallons is a total of 512 portions

To solve this problem, we first need to convert the volume of the soup recipe from gallons to fluid ounces, and then divide by the size of each portion to find the total number of portions.

Given that

1 gallon = 128 fluid ounces

Therefore, the recipe yields:

20 gallons x 128 fluid ounces per gallon = 2560 fluid ounces

Now, we can find the number of 5-fluid ounce portions by dividing the total volume by the portion size:

2560 fluid ounces / 5 fluid ounces per portion = 512 portions

Therefore, the soup recipe yields 512 portions of 5 fluid ounces each.

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Answer:

18.67

Step-by-step explanation:

my calculate ratio results

The width of the top of the bookcase should be approximately 12.2 inches to give Bria's soap carving collection an area of 300 square inches.

Simply multiply the length and width measurements to get the area of your square or rectangular area in square inches.

Assume the bookcase's top is rectangular, with length "L" and width "b". Because the area of a rectangle is the product of its length and width, we get:

L * b = 300

To find "b," we can rearrange the equation as follows:

b = 300 / L

We don't have enough information to directly solve for "L," but we can make an educated guess based on the figure provided. Based on the illustration, the bookcase appears to be roughly twice as long as it is wide. So let us suppose:

L ≈ 2b

When we plug this into the equation above, we get:

b = 300 / (2b) (2b)

To simplify, we have:

b^2 = 150

When we take the square root of both sides, we get:

b ≈ 12.2

We have, rounded to the nearest tenth of an inch:

b ≈ 12.2 inches

As a result, the width of the top of the bookcase should be approximately 12.2 inches in order to give Bria's soap carving collection a 300-square-inch area.

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The derivative of the function g(t) =  -2tsint - sint + 2tcost.

In mathematics, the derivative of a function of a real variable measures the sensitivity of the function's value (the output value) to changes in its independent variable (the input value). Derivatives are a fundamental tool of calculus. The process of finding derivatives is called differentiation. The reverse process is called retro differentiation.

The fundamental theorem of calculus associates inverse differentiation with integration. Differentiation and integration form the two basic operations in univariate calculus.

Given that:

g(t) = tan(ln(t)) g'(t)

Now, Differentiating for the first time:

  (t²+1) d/dt (sin t) + Sin t d/dt (t²+1)

= (t²+1) cos t + sint (2t)

Now,

h'(t) = t²cos t + cost + 2tsint

Differentiating again:

h''(t) = d/dt (t²cos t) + d/dt (cost) + d/dt (2tsint)

      = -2tsint - sint + 2tcost

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The coordinates of p so that p partitions segment ab in the part-to-whole ratio of 1 to 5 with a(-9, 3) and b(1, 8) is (-44/6, 23/6).

To find the coordinates of point P, we can use the following formula:

P = ( (5 x Ax + 1 x Bx) / 6 , (5 x Ay + 1 x By) / 6 )

Here, Ax and Ay are the x and y coordinates of point A, and Bx and By are the x and y coordinates of point B. By plugging in the values we have, we get:

P = ( (5 x (-9) + 1 x 1) / 6 , (5 x 3 + 1 x 8) / 6 )

P = ( (-45 + 1) / 6 , (15 + 8) / 6 )

P = ( -44/6 , 23/6 )

So the coordinates of point P are (-44/6, 23/6).

To check if we got the right answer, we can measure the distance between A and P and the distance between P and B, and make sure that the ratio is indeed 1 to 5. We can use the distance formula for this:

Distance between A and P:

√( (-9 - (-44/6))² + (3 - 23/6)² )

= √( (35/6)² + (5/6)² )

= √( 1225/36 + 25/36 )

= √( 1250/36 )

= 5/6 x √(50)

Distance between P and B:

√( (1 - (-44/6))² + (8 - 23/6)² )

= √( (55/6)² + (41/6)² )

= √( 3025/36 + 1681/36 )

= √( 4706/36 )

= 5/6 x √(94)

The ratio of the distance between A and P to the distance between P and B is:

(5/6 x √(50)) / (5/6 x √(94)) = √(50/94)

Simplifying this gives us:

√(50/94) = √(25/47)

The ratio of the distances is indeed 1 to 5, which confirms that our answer for the coordinates of P is correct.

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The sum of the distinct prime factors is 16.

There are overall 24 factors of 792 among which 792 is the most significant factor and its prime factors are 2, 3, and 11.

Hence sum = 2 + 3 + 11 = 16

The value of x is 5

Triangles with the same shape but different sizes are said to be similar triangles. To be more specific, two triangles are comparable if their respective sides are proportionate and their corresponding angles are congruent.

Two triangles are similar if corresponding angles are congruent and corresponding sides are proportional.

from the below figure, both the triangles are similar, ∆ABC ≈ ∆EFB

By using Thales's theorem, the ratio of the sides of triangles are;

BE/EA = BF/FC

15/30 = x/10    

x = 5

Therefore the value of x is 5

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The ratio of triangle sides can be calculated using Thales' theory, and it is the value of x is 5

Similar triangles are those with the same shape but varying sizes. To be more precise, two triangles are comparable if their matching angles and respective sides are congruent.

If matching sides are proportional and corresponding angles are congruent, two triangles are similar.

Both triangles in the following figure are comparable ∆ABC ≈ ∆EFB

The ratio of triangle sides can be calculated using Thales' theory, and it is;

BE/EA = BF/FC

15/30 = x/10    

x = 5

Therefore the value of x is 5

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there is a 20% chance that the point chosen at random will lie on bg.

To find the probability that a point chosen at random will be on the line segment bg, we need to consider the length of bg in relation to the length of the entire line segment ak.

Let us assume that ak is a straight line segment, and bg is a smaller segment that lies entirely within it. To find the probability, we need to divide the length of bg by the length of ak.

Let the length of bg be x and the length of ak be y. Then the probability that a point chosen at random will be on bg is:

Probability = Length of bg / Length of ak

Probability = x / y

However, we need to be careful here. If we choose a point anywhere on ak, it may not necessarily lie on bg. There are an infinite number of points on ak, but only one segment bg. Therefore, the probability we are looking for is actually the ratio of the lengths of bg to ak.

So, if we know the lengths of bg and ak, we can find the probability by dividing them. For example, if bg is 2 units long and ak is 10 units long, the probability of choosing a point on bg is:

Probability = 2 / 10

Probability = 0.2 or 20%

In this case, there is a 20% chance that the point chosen at random will lie on bg.

In conclusion, the probability of a point chosen at random on ak being on bg is directly proportional to the length of bg in relation to the length of ak. Therefore, we need to find the ratio of the lengths of the two line segments to determine the probability.

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what is the probability  a point is chosen at random on ak and then the point will be on bg. dont forget to reduce the products?

If a flood occurs, there is a 72.7%  probability that it rains non-stop in December of that year

What is probability ?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain to occur. Probabilities between 0 and 1 indicate the likelihood of an event occurring, with smaller values indicating lower likelihoods and larger values indicating higher likelihoods.

Probabilities can be determined using mathematical formulas, statistical analysis, and empirical data. They are used in a wide range of fields, including mathematics, science, economics, and social sciences, to make predictions and inform decision-making.

According to the question:
Let's use the following notation:

NR: Non-stop rain

F: Flood

We want to find the conditional probability of NR given that F occurred, or P(NR|F). We can use Bayes' theorem to find this probability:

P(NR|F) = P(F|NR) * P(NR) / P(F)

We know that 50% of the rainy days are non-stop rain, so P(NR) = 0.5. We also know that if there is non-stop rain, 80% will flood, so:

P(F|NR) = 0.8

If there is no non-stop rain, 30% flooding will occur, so:

P(F|~NR) = 0.3

We can use the law of total probability to find P(F), which is the probability of flooding occurring:

P(F) = P(F|NR) * P(NR) + P(F|~NR) * P(~NR)

P(~NR) = 1 - P(NR) = 0.5

P(F) = 0.8 * 0.5 + 0.3 * 0.5 = 0.55

Now we can plug in all the values to find P(NR|F):

P(NR|F) = P(F|NR) * P(NR) / P(F)

P(NR|F) = 0.8 * 0.5 / 0.55 = 0.727 or 72.7% (rounded to one decimal place)

Therefore, if a flood occurs, there is a 72.7% chance that it rains non-stop in December of that year.

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Step 1: Arrange the arrays so that A and B are in the same order: A = [ 1 2 4 0 5 6 ], B = [ 7 3 2 5 1 9]

Step 2: To find C = A+B, add each element of A and B together.

C = [1+7, 2+3, 4+2, 0+5, 5+1, 6+9]

C = [8, 5, 6, 5, 6, 15]

Step 3: To find D = A-B, subtract each element of B from A.

D = [1-7, 2-3, 4-2, 0-5, 5-1, 6-9]

D = [-6, -1, 2, -5, 4, -3]

Step-by-step explanation:

[tex] \frac{1 - \ { \sin(t) }^{2} } { { \sin(t) }^{2} } = \frac{1} { { \sin(t) }^{2} } - \frac{{ \sin(t) }^{2}}{{ \sin(t) }^{2}} \: = \frac{1} { { \sin(t) }^{2} } - 1[/tex]

[tex] = { \sin(t) }^{ - 2} - 1[/tex]

hope this helps

Answer:

1. Volume:   912 cm³

  Surface Area:  672 cm²

2. Volume:  1239 1/3 cm³

  Surface Area:  819 cm²

3.  Volume:  216 2/3 cm³

  Surface Area:  240 cm²

Step-by-step explanation:

Formula for Volume of a Pyramid:  [tex]\frac{1}{3} * lwh[/tex]

(where l, w, and h are the length, width, and height, respectively)

Formula for Surface Area of a Square Pyramid:  [tex]B + \frac{1}{2} ps[/tex]

(where B is the area of the base, p is the perimeter of the base, and s is the slant height)

1 (Volume) - V = (1/3)(12)(12)(19)

                       = (4)(12)(19)

                       = 912 cm³

1 (Surface Area) - SA = (12)(12) + (1/2)(4)(12)(22)

                                  = 144 + 528

                                  = 672 cm²

2 (Volume) - V = (1/3)(13)(13)(22)

                        = 1239 1/3 cm³

2 (Surface Area) - SA = (13)(13) + (1/2)(4)(13)(25)

                                   = 169 + 650

                                   = 819 cm²

3 (Volume) - V = (1/3)(10)(10)(6.5)

                        = 216 2/3 cm³

3 (Surface Area) - SA = (10)(10) + (1/2)(4)(10)(7)

                                   = 100 + 140

                                   = 240 cm²

Answer: 62°

Step-by-step explanation:

All angles of a triangle add up to 180°.

So, add up all the other angles.

15 + 25 + 39 = 79

180 - 79 = 101

Then, to find x, subtract 39 from 101, to get 62!

The volume of figure made up of two rectangular prisms is found to be  1092  sq. in.

volume of rectangular prisms = length * width * height

So,

Volume of A and C are same as their dimensions are same.

Volume of A and C = 2*(7*2*3)

Volume of A and C =  84 sq. in

Now, volume of B = 12*12*7

volume of B = 1008 sq. in.

Complete volume = Volume of A and C +  volume of B

Complete volume = 84 sq. in + 1008 sq. in.

Complete volume = 1092  sq. in.

Thus, the volume of figure made up of two rectangular prisms is found to be  1092  sq. in.

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The factorization of the polynomial x³ + 9x² + 3x + 27 by grouping is:

x³ + 9x² + 3x + 27 = (x + 9)(x² + 3)

In algebra, "grouping" refers to a method of factoring polynomials that involves grouping together pairs of terms within the polynomial, in order to factor out a common factor.

In mathematics, a polynomial is a mathematical expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

In the given question,

To factor the four-term polynomial by grouping, we can follow these steps:

Step 1: Group the first two terms and the last two terms together.

x³ + 9x² + 3x + 27

= (x³ + 9x²) + (3x + 27)

Step 2: Factor out the common factor from each group.

= x²(x + 9) + 3(x + 9)

Step 3: Notice that the expression (x + 9) is a common factor of both terms, and factor it out.

= (x + 9)(x² + 3)

Therefore, the factorization of the polynomial x³ + 9x² + 3x + 27 by grouping is:

x³ + 9x² + 3x + 27 = (x + 9)(x² + 3).

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Answer:

Proofs attached to answer

Step-by-step explanation:

Proofs attached to answer

There are 4 terms in the series.

what is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers where each term is equal to the previous term plus a fixed constant difference, called the common difference.

The given series is an arithmetic sequence with a common difference of 16. We can find the first term by plugging in n=1 into the sequence:

a1 = -4 + 16(-1)^1 = -4 + 16 = 12

Using the formula for the sum of an arithmetic sequence, we have:

Sn = n/2(2a1 + (n-1)d)

where n is the number of terms, a1 is the first term, and d is the common difference. Plugging in the given values, we have:

3276 = n/2(2(12) + (n-1)(16))

Simplifying this equation, we get:

3276 = n/2(28 + 16n - 16)

3276 = n/2(12 + 16n)

Multiplying both sides by 2 and rearranging, we get:

6544 = n(6 + 8n)

Dividing both sides by 2, we get:

3272 = n(3 + 4n)

We can see that n must be an even number, since the left side is even. We can also see that n cannot be too large, since the right side increases much faster than the left side. Trying some even values of n, we find that:

n=8 -> 3272 = 8(3 + 4(8)) is false

n=6 -> 3272 = 6(3 + 4(6)) is false

n=4 -> 3272 = 4(3 + 4(4)) is true

Therefore, there are 4 terms in the series.

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Answer:30 inches

Step-by-step explanation: