3) ____ is the expression,
which tells the nature of the roots of a quadratic equation of the form
3) ____ is the expression,
which tells the nature of the roots of a quadratic equation of the form

The resulting three-dimensional figure is a cone with base radius of 17 inches.

A cone is a three-dimensional geometric shape that has a circular base and a curved surface that tapers to a point, called the apex. The base of the cone is a circle, and the height of the cone is the perpendicular distance from the apex to the center of the base. The surface of the cone is made up of all the points that are equidistant from the apex and lie on the base.

Consider an isosceles triangle with 82 inches of circumference and a leg length of 24 inches.

This means that 82 - 2(24) = 82 - 48 = 34 inches is the length of the other side.

The triangle is rotated along a line that passes through points B and D, with point D serving as the intersection of lines AC.

When this happens, a cone will be created in three dimensions with a base radius equal to the length of line AD (i.e. half of the length of line AC being the other side of the isosceles triangle).

The resulting cone's base radius is 1/2 (34) = 17 inches.

Hence, the resulting three-dimensional figure is a cone with base radius of 17 inches.

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(a) The general anti-gradient for ∇f(x,y) = (6x + 4y^3, –2 + 12xy^2) is f(x,y) = 3x^2 + 4xy^3 – 2y + C

(b) The general anti-gradient for ∇f(x,y) = (6x/y^2 -2e^-2x, 6x^2/y^3+4y) is f(x,y) = 3x^2/y^2 + e^-2x + 2y^2 + C

(a) To find the general anti-gradient f for ∇f(x,y) = (6x + 4y^3, –2 + 12xy^2), we need to integrate each component of the gradient with respect to its corresponding variable.

Thus, we have

f(x,y) = ∫(6x + 4y^3) dx + C1

f(x,y) = 3x^2 + 4xy^3 + C1

f(x,y) = ∫(–2 + 12xy^2) dy + C2

f(x,y) = –2y + 4x y^3 + C2

where C1 and C2 are constants of integration

Therefore, the general anti-gradient for ∇f(x,y) = (6x + 4y^3, –2 + 12xy^2) is

f(x,y) = 3x^2 + 4xy^3 – 2y + C

where C is an arbitrary constant.

(b) To find the general anti-gradient f for ∇f(x,y) = (6x/y^2 -2e^-2x, 6x^2/y^3+4y), we need to integrate each component of the gradient with respect to its corresponding variable

Thus, we have

f(x,y) = ∫(6x/y^2 -2e^-2x) dx + C1

f(x,y) = 3x^2/y^2 + e^-2x + C1

f(x,y) = ∫(6x^2/y^3+4y) dy + C2

f(x,y) = 3x^2/y^2 + 2y^2 + C2

where C1 and C2 are constants of integration.

Therefore, the general anti-gradient for ∇f(x,y) = (6x/y^2 -2e^-2x, 6x^2/y^3+4y) is

f(x,y) = 3x^2/y^2 + e^-2x + 2y^2 + C

where C is an arbitrary constant.

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The given question is incomplete, the complete question is:

Find the general anti-gradient, f, for the following: (i.e. f was the original function, and you are given the gradient of the function.) (a)  ∇f(x,y) = (6x + 4y^3, –2 + 12xy^2) (b)  ∇f(x,y) = (6x/y^2 -2e^-2x, 6x^2/y^3+4y)

Answer:

1.5  2.4  3.8

Step-by-step explanation:

1. 5

2. 4

3. 8

4. 6

5. 8

6. 15

7. 8

8. 12

9. 6

(a) There are 20 ways to choose 2 colors without replacement when the order is relevant. (b) There are 10 ways to choose 2 colors without replacement when the order is not relevant.

(a) If the order of the choices is relevant, then we need to use the permutation formula. The number of ways to choose 2 colors from 5, without replacement and with order considered, is given by P(5,2) = 5!/(5-2)! = 5x4 = 20.

(b) If the order of the choices is not relevant, then we need to use the combination formula. The number of ways to choose 2 colors from 5, without replacement and with order not considered, is given by C(5,2) = 5!/(2!3!) = 10.

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The cost of the fat-free cottage cheese required in the recipe is R17.98.

How to determine the cost ?

To determine the cost, we first need to calculate the amount of fat-free cottage cheese required in the recipe. The recipe calls for 250g of fat-free cottage cheese, which can be obtained by using 2 units of 125g each. Knowing the cost of ingredients is important for Mrs Gilfillan to price the cake appropriately to cover her costs and make a profit.

What is the cost of the fat-free cottage cheese required to make the Mixed Berry and Almond Polenta Cake recipe that serves 15 espresso cups at Mrs Gilfillan's coffee shop?

The cost of 1 unit of 125g fat-free cottage cheese is R8.99.

Therefore, the cost of 2 units of 125g is calculated as:

R (2 x 8.99) =R 17.98

Hence, the cost of the fat-free cottage cheese required in the recipe is R 17.98.

This cost is in addition to the cost of the other ingredients, such as butter, sugar, ground almonds, mixed frozen berries, and polenta, as well as the cost of labor, overhead, and other expenses involved in making and selling the cake.

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The perimeter of the figure is 24 feet.

The perimeter of a shape is the distance between its edges. Learn how to calculate the perimeter of various shapes by adding the side lengths.

To calculate the perimeter of the figure, add the lengths of all the sides. To calculate the perimeter of the figure, add the lengths of all the sides. Label the diagram's points as follows:

Beginning at the top left and working our way clockwise, we have:

A at (0, 4)

B at (6, 4)

C at (6, 0)

D at (4, 0)

E at (4, 2)

F at (2, 2)

G at (2, 0)

H at (0, 0)

Using the distance formula, we can now calculate the length of each side:

[tex]AB = \sqrt{((6-0)^2 + (4-4)^2)}= 6 feet[/tex]

[tex]BC = \sqrt{((6-6)^2 + (0-4)^2)} = 4 feet[/tex]

[tex]CD = \sqrt{((4-6)^2 + (0-0)^2)} = 2 feet[/tex]

[tex]DE = \sqrt{((4-4)^2 + (2-0)^2)} = 2 feet[/tex]

[tex]EF = \sqrt{((2-4)^2 + (2-2)^2)} = 2 feet[/tex]

[tex]FG = \sqrt{((2-2)^2 + (0-2)^2)} = 2 feet[/tex]

[tex]GH = \sqrt{((0-2)^2 + (0-0)^2) }= 2 feet[/tex]

[tex]HA = \sqrt{((0-0)^2 + (4-0)^2)}= 4 feet[/tex]

The perimeter is equal to the sum of these lengths:

Perimeter = 6 + 4 + 2 + 2 + 2 + 2 + 2 + 4

= 24 feet

As a result, the figure's perimeter is 24 feet.

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The homotopy classes of paths from x to x in a space X refer to a set of equivalence classes of continuous paths that start and end at the same point, x, in the space X, where equivalence is defined in terms of homotopy.

In other words, for any two paths, there exists a continuous transformation (called a homotopy) between them such that the endpoints remain fixed. Two paths are said to be homotopic if they can be continuously deformed into each other while keeping their endpoints fixed. The set of all paths that are homotopic to each other forms an equivalence class.

The homotopy classes of paths from x to x are important in algebraic topology, as they provide a way to study the topological structure of a space by analyzing the properties of the paths within it. They can also be used to define higher algebraic structures such as the fundamental group and higher homotopy groups.

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Profit ratios are used to assess a company's profitability by evaluating its profit in relation to other financial indicators like sales, assets, or equity.

The return on assets ratio is used to evaluate the connection between the assets your business uses and the profits it makes. You calculate it using information from the balance sheet and the income statement. (The ratio is multiplied by 100 to become a percentage.)

But, this does not necessarily imply that this is the profit margin you should aim for, a modest margin is 5%. A healthy margin is 10%, and a large margin is 20%.

Therefore, To evaluate job performance using these methods, you would need to define your research goals and variables, select your cross-section and time period, collect and analyse the relevant data, and draw conclusions based on your findings.

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The piecewise function will be continuous in its domain if k = 49/48

Here we have a piecewise function and we want to find the value of k suc that the function is continuous on the domain.

To get that, both piecese of the function must ahve the same value in the jump, then we will get:

f(7) = f(7)

Replacing the functions we will get:

k·7²  = 7·7 + k

Let's solve that equation for k.

49k = 49 + k

49k - k = 49

48k = 49

k = 49/48

That must be the value of k such that the function is continuous in all its domain.

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Answer: The range is 20, Standard Deviation, σ: 6.8702256149271

Step-by-step explanation:

I hope it helped!

<3

At equilibrium we expect [Reactants} < [Products)  from the the profile that has been shown.

An energy profile diagram, also known as an energy diagram or reaction coordinate diagram, is a graphical representation of the energy changes that occur during a chemical reaction or a physical process. It shows the energy levels of the reactants, products, and any intermediate species that may form during the reaction.

The horizontal axis of an energy profile diagram represents the reaction coordinate, which is a measure of the progress of the reaction from the reactants to the products. The vertical axis represents the energy of the system.

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The formula to calculate the percentage of the Sticker Price in column E using structured references would be:

=IFERROR((1-[Sale Price]/[Sticker Price])*100,"% Discount")

A number or quantity can be expressed as a fraction of 100 using a percentage.It is often denoted by the symbol % (percent).

Assuming that the Sticker Price is located in column C and the Sale Price is located in column D, the formula to calculate the percentage of the Sticker Price in column E using structured references would be:

=IFERROR((1-[Sale Price]/[Sticker Price])*100,"% Discount")

In this formula, the percentage of the Sticker Price is calculated by subtracting the Sale Price from the Sticker Price, dividing the difference by the Sticker Price, and then multiply by 100 to convert the result into percentage. The IFERROR function is used to handle any errors that may occur if the Sale Price is missing or is equal to or greater than the Sticker Price. If an error occurs, the formula will display the text "% Discount".

Note that the structured references [Sale Price] and [Sticker Price] refer to the column headers rather than the cell references. This makes the formula more dynamic and easier to read, especially when working with large datasets.

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Answer:

159 proper subsets.

Step-by-step explanation:

Given a set {2, 4, 6, 8...100}, how many proper subsets are there?

First, find how many subsets there are in 2 - 10:

That's 16.

Then because there are 10 10s in 100, multiply by 10:

16 x 10 = 160

Finally, because it says proper subsets, subtract by 1:

160 - 1 = 159 proper subsets.

Therefore, there are 159 proper subsets in {2, 4, 6, 8...100}

Explanation: The evolution of numbers developed differently with disparate versions, which include the Egyptian, Babylonians, Hindu-Arabic, Mayans, Romans, and the modern American number systems.

Answer:

the difference is the first one, (-3)^2, is negative while the other is positive

Step-by-step explanation:

Answer:

p (even,odd)- more likely

p(2,5) - less likely

p(odd,1)- less likely

Answer:

If the time is 3:45 how many minutes is it slow or fast

Answer:

m^2-m-6

Step-by-step explanation:

(m-3)(m+2)

m^2 +2m-3m-6

m^2-m-6

Answer:

m² - m - 6

Step-by-step explanation:

(m - 3)(m + 2)

each term in the second factor is multiplied by each term in the first factor, that is

m(m + 2) - 3(m + 2) ← distribute parenthesis

= m² + 2m - 3m - 6 ← collect like terms

= m² - m - 6

The real solutions of the following equation are 5 and 1

[ Graph is attached below ]

To solve the given equation graphically we need to find the coordinate points that pass through the graph.

This can be done by taking 'x' values and solving for them 'y' values.  

Now draw the graph using the above coordinates and find the solution as shown below.

Here we have

x³ - 6x²+ 5x = 0  

Let y = x³ - 6x²+ 5x  

To draw the graph find the coordinates of points as follows

At x = 0 =>  y = (0) + (0) + (0) = 0

At x = 1 => y = (1)³ - 6(1)² + 5(1) = 0

At x = -1 => y = (-1)³ - 6(-1)² + 5(-1) = - 12

At x = 2 => y = (2)³ - 6(2) + 5(2) = 6

From the above calculation,

The coordinates of the points to draw the graph are (0, 0), (1, 0), (-1, -12), and (2, 6)

Here the solutions of the graph are the x-coordinate of points where the graph cuts the x-axis

From the figure, the graph will cut the x-axis at 1 and 5  

Therefore,

The real solutions of the following equation are 5 and 1

[ Graph is attached below ]

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[tex]80[/tex] billion dollars' worth of net exports were made by China and Germany. The first claim is accurate.

Export is the process of supplying goods and services to some other nation. Contrarily, importing is the act of acquiring goods from outside and transferring them into one's own nation.

The domestic product (GDP) is a measure of all the products and services generated in the United States; thus, changes in exports change significantly in the demand for goods and services made in the United States abroad.

The total of China's and Germany's net exports would be:

[tex]50[/tex] billion + [tex]30[/tex] billion [tex]= 80[/tex] billion

As a result, Germany & China's consolidated net exports amounted to [tex]80[/tex] billion u.s. dollars, reflecting answer option (a).

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Using trigonometric ratio it is proved that sin(B + 2C) + sin(C + 2A) + sin(A + 2B) = 4sin(A-B)/2.cos(B-C)/2.cos(C-A)/2 when A + B + C = 180.

What is trigonometric ratio?

Triangle side length ratios are known as trigonometric ratios. In trigonometry, these ratios show how the ratio of a right triangle's sides to each angle. Sine, cosine, and tangent ratios are the three fundamental trigonometric ratios.

We can start by using the sine addition formula to expand each of the sine terms in the left-hand side of the equation -

sin(B + 2C) = sin(B)cos(2C) + cos(B)sin(2C) = 2sin(B)cos(C)²

sin(C + 2A) = sin(C)cos(2A) + cos(C)sin(2A) = 2sin(C)cos(A)²

sin(A + 2B) = sin(A)cos(2B) + cos(A)sin(2B) = 2sin(A)cos(B)²

Substituting these expressions into the left-hand side of the equation, we get -

2sin(B)cos(C)² + 2sin(C)cos(A)² + 2sin(A)cos(B)²

Factoring out the 2, we can rewrite this as -

2(sin(B)cos(C)² + sin(C)cos(A)² + sin(A)cos(B)²)

Using the trigonometric ratio identity sin(2x) = 2sin(x)cos(x), we can rewrite each of the cosine squared terms as a product of sines and cosines -

cos(C)² = (1/2)(1 + cos(2C)) = (1/2)(1 + 2cos(C)sin(C))

cos(A)² = (1/2)(1 + cos(2A)) = (1/2)(1 + 2cos(A)sin(A))

cos(B)² = (1/2)(1 + cos(2B)) = (1/2)(1 + 2cos(B)sin(B))

Substituting these expressions into the previous equation, we get -

2(sin(B)(1/2)(1 + 2cos(C)sin(C)) + sin(C)(1/2)(1 + 2cos(A)sin(A)) + sin(A)(1/2)(1 + 2cos(B)sin(B)))

Simplifying and grouping the terms, we get -

sin(B)sin(C)cos(C) + sin(C)sin(A)cos(A) + sin(A)sin(B)cos(B)

Using the sine addition formula again, we can rewrite each of the cosine terms as a product of sines -

cos(C) = sin(A + B)

cos(A) = sin(B + C)

cos(B) = sin(C + A)

Substituting these expressions into the previous equation, we get -

sin(B)sin(C)sin(A + B) + sin(C)sin(A)sin(B + C) + sin(A)sin(B)sin(C + A)

We can rearrange this expression by factoring out a sin(A-B)/2 sin(B-C)/2 sin(C-A)/2 term -

sin(A-B)/2 sin(B-C)/2 sin(C-A)/2 (cos(A) - cos(B) + cos(B) - cos(C) + cos(C) - cos(A))

Simplifying the terms in parentheses, we get -

sin(A-B)/2 sin(B-C)/2 sin(C-A)/2 (0)

Therefore, the left-hand side of the equation simplifies to 0, which is equal to the right-hand side of the equation -

4sin(A-B)/2.cos(B-C)/2.cos(C-A)/2

Therefore, we have proven that sin(B + 2C) + sin(C + 2A) + sin(A + 2B) = 4sin(A-B)/2.cos(B-C)/2.cos(C-A)/2.

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Random sampling is crucial when surveying as it ensures that the sample selected is representative of the population.

By randomly selecting participants from the population, the sample is likely to be unbiased on average, which means that the results of the study can be generalized to the entire population. Without random sampling, the results of the study may be skewed or biased towards a certain group, which can lead to incorrect conclusions and poor decision-making. Therefore, it is essential to use random sampling when surveying to obtain accurate and reliable results.

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Answer:

x=9

Step-by-step explanation:

this is solved by using cover and cancel so we can isolate the x variable

1) first add 72 to both sides.

6x=54

2) second divide both sides by 6

3) x=9

For the triangles to be similar the value of FJ = 9.

Triangles that resemble one another but may not be precisely the same size are said to be comparable triangles. When two items have the same shape but different sizes, they can be considered to be comparable. This indicates that comparable forms superimpose one another when amplified or demagnified. The term "Similarity" refers to this characteristic of like forms.

Let us suppose the value of FJ = x.

Now, for the triangle that are similar the corresponding sides are in proportion.

That is,

FG/FH = FJ/FG

Here, FH = x + 27

Substituting the values we have:

18/x + 27 = x / 18

18(18) = x (x + 27)

x² + 27x - 324 = 0

Splitting the middle term we have:

(x + 36) (x - 9) = 0

x = -36 and x = 9

Since, length cannot be negative we have the value of x = 9.

Hence, for the triangles to be similar the value of FJ = 9.

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a) The maximum value for this data set is 31, the minimum value is 11 and the median value is 15.5.

b) Q1 13.5

Q2 15.5

Q3 24

c) 10.5

d) lower than 11 or higher than 31

e) Yes.

f) This data set is slightly positively skewed.

The median is the middle value of a set of data when the values are arranged in numerical order. It is used to find the middle value of a set of numbers or observations.

A. The values can be plotted on the boxplot as follows:

Maximum: 31

Minimum: 11

Median: 15.5

B. To find Q1, Q2 and Q3, the data must first be sorted from lowest to highest: 11; 13; 14; 15; 15; 16; 21; 26; 30; 31. Q1 is the median of the lower half of the data set (11, 13, 14, 15 and 15), which is 13.5. Q2 is the median of the entire data set (15.5). Q3 is the median of the upper half of the data set (16, 21, 26, 30 and 31), which is 24. The values can be plotted on the boxplot as follows:

Q1: 13.5

Q2: 15.5

Q3: 24

C. The Interquartile Range (IQR) is calculated by subtracting Q1 from Q3: IQR = Q3 - Q1 = 24 - 13.5 = 10.5.

D. Outliers are values that are significantly greater or lower than the majority of the data set. In this data set, values that are lower than 11 or higher than 31 would be considered outliers.

E. Yes, this data set has outliers. The value 11 is significantly lower than the majority of the data set, and the value 31 is significantly higher than the majority of the data set.

F. This data set is slightly positively skewed, as the majority of the data points are clustered around the lower end of the range, and there is a longer tail of higher values. This is indicated by the fact that the median (15.5) is lower than the mean (19.4).

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a) The maximum value of the given dataset is 31, while the minimum value is 11 and the median is 15.5.

b) Q1 = 14,

Q2 = 15.5,

Q3 = 21

c) (IQR) = 7

d) Any values below 11 or above 31

e) Yes

f) slightly positively skewed.

Median is a measure of central tendency that is calculated by taking the middle value of a set of numbers when the values are arranged in ascending or descending order.

a) Maximum value: 31,

Minimum value: 11,

Median value: 15.5

b) Q1 = 14,

Q2 = 15.5,

Q3 = 21

c) The Interquartile Range (IQR) is calculated by subtracting Q1 from Q3:

Inter Quartile Range (IQR) = Q3-Q1

=21-14

= 7

d) Any values below 11 or above 31 can be considered as outliers.

As 11 is the lowest value given and 31 is the highest value.

e) Yes, this data set has outliers. The values 31 and 11 are both outliers since they are beyond the range of the Inter Quartile Range (IQR).

The value 11 is significantly lower than the majority of the data set, and the value 31 is significantly higher than the majority of the data set.

f) This data set is slightly positively skewed.

As the majority of the data points are clustered around the lower end of the range, and there is a longer tail of higher values.

This is indicated by the fact that the median (15.5) is lower than the mean.

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Answer:

To calculate the monthly installment for Justin's car loan, we can use the formula for the present value of an annuity:

PV = (PMT / r) x [1 - 1 / (1 + r)^n]

where PV is the present value of the loan, PMT is the monthly payment, r is the interest rate per period (in this case, per month), and n is the number of periods (in this case, the number of months in 4 years, which is 48).

We know that PV is the amount of the car loan, which is N$18500. We also know that r is 6% per year, which is equivalent to 0.5% per month (since there are 12 months in a year). Finally, we know that n is 48.

Substituting these values into the formula, we get:

18500 = (PMT / 0.005) x [1 - 1 / (1 + 0.005)^48]

Simplifying this expression:

18500 = (PMT / 0.005) x 36.4358

PMT = 18500 / 36.4358 / 0.005

PMT ≈ N$402.87

Therefore, Justin's monthly installment for his car loan is approximately N$402.87.

Check the picture below.

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We have proved that quadrilateral ABCD is a square below.

The perimeter of ABCD = 4 × AB

The area of ABCD= (AB)² .

A square is a two-dimensional, four-sided shape with all sides of equal length and all angles of 90 degrees. I

Proof that Quadrilateral ABCD is a Square:

We can prove that quadrilateral ABCD is a square by using the properties of a square.

A square is also a quadrilateral with four equal sides and four right angles.

In the image, we can see that side AB = side CD, and side AD = side BC.

Furthermore, we can see that the angles A, B, C, and D are all right angles=90 degree

This means that the quadrilateral ABCD is a square.

Finding the Perimeter of ABCD:

The perimeter of a square is equal to the sum of its four sides.

In the image, we can see that the sides AB, CD, AD, and BC are all equal. Therefore, the perimeter of ABCD=

4 × AB

= 4 × CD

= 4 × AD

= 4 × BC.

Finding the Area of ABCD:

The area of a square is equal to the length of one of its sides squared. In the image, we can see that the sides AB, CD, AD, and BC are all equal. Therefore, the area of ABCD= (AB)²

= (CD)²

= (AD)²

= (BC)².

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We have proved that quadrilateral ABCD is a square below.

Perimeter of ABCD is 4 × AB

The area of ABCD= (AB)².

A square is a two-dimensional, four-sided shape with equal-length edges and 90-degree angles on each side.

The quadrilateral ABCD is a square, therefore:

Using the characteristics of a square, we can demonstrate that the parallelogram ABCD is a square.

A quadrilateral with four equal edges and four right angles is a square.

As seen in the picture, side AB corresponds to side CD and side AD corresponds to side BC.

In addition, we can see that the orientations A, B, C, and D are all 90-degree right angles.

The parallelogram ABCD is a square as a result.

Finding ABCD's Perimeter:

The sum of a square's four edges determines its perimeter.

The sides AB, CD, AD, and BC are equal on the picture, as can be seen. Consequently, the ABCD boundary

4 × AB

= 4 × CD

= 4 × AD

= 4 × BC

Finding the Area of ABCD:

A square's area is equivalent to the square of one of its sides. The sides AB, CD, AD, and BC are equal on the picture, as can be seen.

Consequently, ABCD's area equals (AB)².

= (CD)²

= (AD)²

= (BC)².

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The clear image of quadrilateral attached below,