what is square root of 727878

Answer:

  b.  40

Step-by-step explanation:

You want the missing leg length in a right triangle with one side of length 9 and the hypotenuse of length 41.

The missing side of any triangle will be between the sum and difference of the other two side lengths.

  c-a < b < c+a

  32 < b < 50

In a right triangle, every side length is less than the length of the hypotenuse:

  b < c

  32 < b < 41

There is only one answer choice in this range: b = 40.

__

Additional comment

You can make use of your knowledge of Pythagorean triples. One of them is {9, 40, 41}, which tells you that b=40.

Or, you can use the Pythagorean theorem to compute b:

  a² +b² = c²

  9² +b² = 41²

  b² = 1681 -81 = 1600

  b = √1600 = 40

For any odd integer, there is a Pythagorean triple with sides 1 unit apart such that those longer side lengths are about half of the square of that odd integer:

  9² = 81 = 40+41

so {9, 40, 41} is a Pythagorean triple.

The simplest of these is ...

  3² = 9 = 4+5   ⇒   {3, 4, 5} is a Pythagorean triple.

From the given data, 6 employees are ill and 7 are not ill after ate in cafeteria. and 1 employee is ill and 25 are not ill after did not ate in cafeteria.

In the given question,

Results of the questionnaire were obtained for 39 of the 40 workers at the plant.

Six of the seven ill employees ate at the factory cafeteria on September 26 and 27.

7 of 32 non-stricken workers who ate at the factory cafeteria in both days.

We have to construct a 2x2 table using these data.

6 employees are ill and 7 are not ill after ate in cafeteria.

1 employee is ill and 25 are not ill after did not ate in cafeteria.

So the table should be

                                          ILL                    Not ILL                Total

Ate in Cafeteria                  6                          7                         13

Did not ate in cafeteria       1                          25                      26

TOTAL                                  7                         32                       39

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The right question is

Results of the questionnaire were obtained for 39 of the 40 workers at the plant. Six of the seven ill employees ate at the factory cafeteria on September 26 and 27, compared to 7 of 32 non-stricken workers who ate at the factory cafeteria in both days. You also learn that the factory cook died from respiratory complication 3 weeks after hospital admissions.

In the preliminary analysis of results that you will provide to the Health Commissioner, you set up a 2 x 2 table, relating exposure and disease. You calculate the attack rate, case-fatality rate, and the relative risk of getting botulism after eating at the factory cafeteria on September 26 and 27.

1) Construct a 2x2 table using these data.

                                          ILL                    Not ILL                Total

Ate in Cafeteria

Did not ate in cafeteria

TOTAL                                                                                         39

The area of the triangle with the vertices (0, 5), (2, -2), and (5, 1) by using the calculus is  21 square unit.

We need to find the equation among all possible pairs and then integrate the equations from one co-ordinate to another co-ordinate

Equation of line passing through (0,5) and (2,-2) is

y-5 = [(-2-5)/(2-0)](x-0)

=>y-5 = (-7) /2x

=>y= (-7/2x)+5 -------(eq1)

Equation of line passing through (0,5) and (5,1) is

y-5 = [(1-5) / (5-0)](x-0)

=>y-5 = (-4/5)x

=>y = (-4/5)x+5-------(eq2)

Equation of line passing from (2,-2) and (5,1) is

y-(-2) = [[1 - (-2)] / (5-2)] / (x-2)

=>y+2=(3/3)(x-2)

=>y=x-4--------(eq3)

Now, we use definite integration to find the area between the different equation of line.

So, area enclosed between the equations is given by the

area =[tex]\int\limits^5_2[/tex][(-4/5)x+5 - (-7/2)x + 5)dx  + [tex]\int\limits^5_1[/tex][(-4/5)x+5 -(x-4)]dx

=>area=[tex]\int\limits^5_2[/tex](7/2-4/5)x dx + [tex]\int\limits^5_1[/tex]((-9/5)x+9)dx

Using properties of integration,[tex]\int\limits x\, dx=x^{2}/2[/tex]

=>area=[tex]\int\limits^5_2[/tex](27/10)x dx + [tex]\int\limits^5_1[/tex](-9/5)x+9)dx

=>area=([27/10)×[5² - 2²])/2 + [ (-9/5)×(5²-1²) ]/2 +9×(5-1)

=>area=(27/20)×(25-4) + (-9/5)×24+9×4

=>area = (27×21)/20 + (-216)/5+ 36

=>area=(567/20) - (216/5) + 36

=>area= [(567-261×4)+(36×20)]/20

=>area=[(567-864)+720]/20

=>area=423/20

=>area=21 square unit.

Hence, area of triangle is 21 square unit.

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A wheel is an example of rotational symmetry . Many shapes such as squares, circles, and regular hexagons have rotational symmetry. As wheel is a rotational shape thing that's why it is consider as an example of rotational symmetry.

If the image is rotated 120 degrees of hers, then another 120 degrees of his, and finally 120 degrees of his third his, it will look the same in all three positions. This property is called rotational symmetry. Rotational symmetry is found, for example, in circles, squares, rectangles, and all regular polygons. Take an object and rotate it up to 180 degrees around its center. An object has rotational symmetry if it looked the same at any time before the revolution.

Center:

The center, fulcrum, or center of balance refers to a specific point at which an object can spin, spin, or rotate.  When we find this position, we have found the center of the object. By rotating around this center, we can determine the rotational symmetry of the rectangles.

Rotational Symmetry and Order:

Rotational symmetry is a measure of how often an object appears to rotate 360 ​​degrees in the same way. If it matches the original shape only twice, it has degree 2. If it matches 3 times, then the order is 3, and so on. Common rotation orders and object rotation amounts are: Order 2 is 180 degrees, order 3 is 120 degrees, and order 4 is 90 degrees. Order No. 5 has a temperature of 72 degrees, Order No. 6 has a temperature of 60 degrees.

Regular polygons:

Many everyday objects exhibit rotational symmetry. In mathematics, all regular polygons have rotational symmetry. For example, consider an equilateral triangle.  An equilateral triangle corresponds three times as it rotates around its center. As a result, it exhibits cubic rotational symmetry.  

Angle of Rotation:

Knowing the number of points required to rotate an object about its center is important for determining the degree of rotational symmetry, but where and when to adjust it to its original position. I know I should rotate it. In general, if you use an item frequently, physically rotating it will indicate the degree of rotation. Imagine a rectangular mobile phone.  As a result, the rectangular shape of the phone has been rotated 180 degrees twice.

                  degree of rotation = 360°/degree of rotation

You can also calculate the degree by dividing the degree of rotation by: degree = 360°/degree of rotation

A starfish rotates 72 degrees and has an order of 5. The rotational symmetry of the chemical structure of the benzene molecule is of interest. This molecule has rotational symmetry of order 6 with a rotation of 60 degrees.

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The determinant of the given elementary matrices is  20 .

What is a Matrix ?

The matrix is defined as an arrangement of set of numbers arranged in rows and columns to form a rectangular array.

In the matrix the horizontal elements are called as Rows and

the vertical elements are called as Columns .

In the question ,

it is given that ,

the matrix is  [tex]\left[\begin{array}{ccc}8&5\\4&5\end{array}\right][/tex]  ,

the determinant of the matrix is represented as ⇒ (8×5 - 5×4)

Simplifying the given expression further ,

we get ,

= 40 - 20

= 20 .

Therefore , the determinant is 20 .

The given question is incomplete, the complete question is

Compute the determinants of the elementary matrices given as [tex]\left[\begin{array}{ccc}8&5\\4&5\end{array}\right][/tex] .

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Using the perimeter and area formular for a rectangle and semicircle, the dimension of the window should have a length of 1.68 ft and the width should also be 1.68 ft.

Dimension is measurement of width or something.

Side lengths of the window = L

Radius of semicircle = r

width = 2r

Perimeter of semicircle = πr + 2r

Hence, perimeter of window can be expressed thus :

2L + 2r + πr = 12 - - - - (1)

Area of window :

2rL + 0.5πr^2 - - - - - (2)

A = r(12 - 2r - πr) + 0.5πr^2

A = 12r - 2r^2 - πr^2 + 0.5πr^2

A = 12r - 2r^2 - 0.5πr^2

Take the derivative of A to obtain maximum value of r

dA/dr = 12 - 4r - πr = 0

4r + πr = 12

2L = 12 - 2r - πr

12 - 2*1.68 - 1.68(3.14)

2L = 3.3648

L = 3.3648 / 2

L = 1.68.

Hence, the dimension of the window should be : 1.68 by 1.68.

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Answer:

---------------------------------

Total number of hours is w + l and it should be at least 17 hours, (equal to 17 or greater).

We can set inequality as:

For given study the hours of sleep per night is independent of the age.

Also, estimate of the percentages of individuals

                           Hours of sleep

Age       <6      6-6.9       7-7.9         ≥8         Total

≤ 39       38        60           75            67          240

≥ 40       36        57           73            94          260

Total      74        117         148          161        500

%       14.8%   23.4%     29.6%      32.2%

Let the null hypothesis H0 : Hours of sleep per night is independent of age

alternate hypothesis Ha : Hours of sleep per night is not independent of age

Expected frequencies :

E1: ((38+60+75+67)*(38+36))/500 = 35.52

E2: ((38+60+75+67)*(60+57))/500 = 56.16

E3: ((38+60+75+67)*(77+75))/500 = 72.96

E4: ((38+60+75+67)*(65+92))/500 = 75.36

E5: ((36+57+73+94)*(38+36))/500 = 38.48

E6: ((36+57+73+94)*(60+57))/500 = 60.84

E7: ((36+57+73+94)*(77+75))/500 = 79.04

E8: ((36+57+73+94)*(65+92))/500 = 81.64

Then performing the Chi-square independence test X²

X² = Σ[(f1-e1)²/e1 + (f2-e2)²/e2... (fn - en)²/en]

F1 to f8 = 38,60,75,67,36,57,73,94

X² = ((38-49.92)^2)/35.52 + ((60-56.16)^2)/56.16 + ((77-72.96)^2)/72.96 + ((65-75.36)^2)/75.36 + ((36-38.48)^2)/38.48 + ((57-60.84)^2)/60.84 + ((75-79.04)^2)/79.04 + ((92-81.64)^2)/81.64

X² = 4.00

α = 0.05

Reject H0, if p < α

df = (row - 1)*( column - 1)

df = (2 - 1)*(4-1)

   = 3

To find the p value from the X² square score at α = 0.05 and 3 degree of freedom

p = 0.261

Since, p > α ( 0.261 > 0.05) ; The result is not significant at α = 0.05 ; Hence we fail to reject the H0 (null hypothesis).

Now we need to estimate of the percentages of individuals who sleep fewer than 6 hours, 6 to 6.9 hours, 7 to 7.9 hours, and 8 hours or more per night.

                          Hours of sleep

Age       <6      6-6.9       7-7.9         ≥8         Total

≤ 39       38        60           75            67          240

≥ 40       36        57           73            94          260

Total      74        117         148          161        500

%       14.8%   23.4%     29.6%      32.2%

Therefore, we can conclude that the hours of sleep per night is independent of the age.

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In the given Fibonacci sequence , 8th number is 13 mad 11th number is 55 .

What is a Fibonacci Sequence?

The Fibonacci Sequence is the series of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

The next number is found by adding up the two numbers before it:

In a Fibonacci sequence, the number is equal to the sum of the two preceding numbers.

so the seventh number is 3+5 = 8

8th number is 5+ 8 = 13

11th number is 21+34 = 55

now the sequence becomes 0,1,1,2,3,5,8,13,21,34,55

Hence , the 8th number is 13 mad 11th number is 55.

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Answer:

y=2x-14

Step-by-step explanation:

Perpendicular lines have slopes that are opposite reciprocals:

Example: 4 and -1/4 are opposite reciprocals

y = -x/2+2  ==> slope for this equation is -1/2

Now take opposite reciprocal of -1/2

1/(-1/2) * (-1) =

1 * (-2/1) * (-1) =

1 * (-2) * (-1) =

1 * 2 = 2

y=mx+b  

-4 = 2(5) + b  ==> plug in y=-4, x=5, and m=2 from (5, -4) and a slope of 2

-4 = 10 + b  ==> solve for b

-4 - 10 = 10 - 10 + b  ==> subtract 10 on both sides to isolate b

b = -14

y=2x+(-14)  ==> plugin -14 for b in y=mx+b

y=2x-14

From the given question, relation set - {(9, 1), (6,2), (3, 3), (6,4), (9,5)} does not represent the function.

In mathematics, a connection between one variable (the independent variable, x) and another variable is defined as an expression (the dependent variable, y).For example - f(x) = 4x + 6

The collection of ordered pairs from the question is presented.

It is evident that the relation set exists.

{(9, 1), (6,2), (3, 3), (6,4), (9,5)}

It not accurately depict the function since for x = 6, [y] has two potential values, none of which is possible.

so that's why relation set - {(9, 1), (6,2), (3, 3), (6,4), (9,5)} can not represent a function.

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The length of the other leg can't be found because the length of hypotenuse can't be smaller than the length of any of the sides.

What is Pythagoras theorem ?

The three sides of a right-angled triangle are explained by Pythagoras' theorem, also referred to as the Pythagorean theorem. The hypotenuse square of a right-angled triangle is equal to the sum of the squares of the other two sides, according to the Pythagoras theorem.

Given, hypotenuse = 2 cm

          one leg = 3 cm

Let the other leg to be ' x'

By Pythagoras theorem :

hypotenuse^2 = base^2 + perpendicular^2

                 2^2  =  3^2+ perpendicular^2

               perpendicular^2 = 4 - 9 = -5 which is not possible.

Here, The length of hypotenuse is smaller than the length of other side which clearly indicates that the given right triangle can't exist.

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The assertion that there exists a normal subgroup N of G such that N is of finite index in G and [N ⊂ H] is true and can be proved if "G" is a group and "H" is a subgroup of finite index.

Given,

The statement, "G" is a group and "H" is a subgroup of finite index.

We have to prove that there exists a normal subgroup N of G such that N is of finite index in G and [N ⊂ H].

Group: A group is a finite or infinite collection of elements and a binary operation (referred to as the group operation) in discrete mathematics that together meet the four essential qualities of closure, associativity, identity, and inverse.

Subgroup: If a subgroup "H" also satisfies the four essential criteria of closure, associativity, identity element, and inverse, it is a subset of a group, let's say "G" [denoted by (H ≤ G)].

Here,

By left multiplication, the group G affects the collection of left co-sets (G/H).

Hence it induces the permutation representation [ρ:G→Sₙ], where

[n = |G : H|].

(A permutation representation is a group homomorphism, it should be noted.)

Now, let (N = kerρ) be the kernel of the homomorphism (ρ). Then [N◃G].

The quotient group G/N is isomorphic to a subgroup of Sn according to the first isomorphism theorem.

In particular, G/N is a finite group, hence the index [G:N] is finite.

Finally, we show that [N ⊂ H].

For any (x ∈ N) = [ker (ρ)], we have [x(gH) = gH] for any [g ∈ G].

In particular we have (xH = H),

Hence (x ∈ H).

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If figure A is a scale image of figure B. The value of x will be 72.

The ratio among comparable dimensions of an object and a model with that object is known as an exponent in algebra. The replica will be larger if the scale factor is a whole number. The duplicate will be lowered if the step size is a fraction.

It is given that, Figure A is a scale image of Figure B.

We have to find the value of x, which is obtained by taking the ratio of the sides.

Since figure A is a scale image of figure B, there is a proper scale factor for the given data which is obtained by the ratio.

24/20=x/60

x=(60×24) / 20

x=72

Thus, if figure A is a scale image of figure B. The value of x will be 72.

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Exponential smoothing of time collection records assigns exponentially lowering weights for latest to oldest observations.

In different words, the older the records, the much less priority (“weight”) the records is given; more moderns records is visible as extra applicable and is assigned extra weight. Smoothing parameters (smoothing constants),denoted through α,  decide the weights for observations.

The simple formulation is:

St = ayt-1 + (1 - a) S-1

Where:

α = the smoothing constant, a fee from zero to 1. When α is near zero, smoothing takes place extra slowly. Following this, the satisfactory fee for α is the only that outcomes withinside the smallest suggest squared error (MSE).and t = time period.

One of the blessings of Exponential smoothing is ,the forecast may be built the use of most effective 3 values.

Only 3 portions of records are required for exponential smoothing methods.

1. It  desires the forecast for the maximum latest time period.

2. It desires the real fee for that point period.

3. It desires the fee of the smoothing constant, a weighting thing that displays the load given to the maximum latest records values.

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Answer:

9 pound

Step-by-step explanation:

We know

The new puppy's weight has increased by 32% since her last vet visit.

100% + 32% = 132%

First, we have to find 1%

12 divided by 132% = 0.09 pound

Then we take

0.09 times 100% = 9 pounds

So, the puppy weighs 9 pounds on her last visit.

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Answer:

The answer to the problem is 9 pounds.

Step-by-step explanation:

100+32=132

12/132=.09

.09x100= 9

I know my explanation wasn't very detailed, but I know sometimes people learn by just seeing it instead of having a long drawn out explanation so I wanted to give you that option incase it works better for you.

The option d is correct for the given problem.

What is test statistic?

A statistic used in statistical hypothesis testing is known as a test statistic. A hypothesis test is typically specified in terms of a test statistic, considered as a numerical summary of a data-set that reduces the data to one value that can be used to perform the hypothesis test.

Given:

The data from an independent-measures research study produce a sample mean difference of 12 points and an estimated standard error of 4 points.

Test statistics = (x bar - μ) / (σ / √n)

                      = 12 / 4

Test statistics = 3

Here the test statistics is 3.

Hence, the option d is correct for the given problem.

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Complete question:

The data from an independent-measures research study produce a sample mean difference of 12 points and an estimated standard error of 4 points. If there are n = 10 scores in each sample, then the value for the t statistic is ____.​

Select one:

a. 2

b. 4

c. 1.63

d. 3

The distance from the point to the parametric line is given as follows:

5 units.

The first step is obtaining the vector from the line L(0, 5 + 3t, 2 + t) to the point P(5,5,2), as follows:

P - L = (5, -3t, -t).

The direction vector of the line is given as follows:

(0, 3, 1).

The dot product of the vector with the direction of the line is of zero, hence:

(5, -3t, -t)(0, 3, 1) = 0

-9t - t = 0

-10t = 0

t = 0.

Thus the point on the line which we will use to obtain the distance is of:

(0, 5 + 3(0), 2 + 0) = (0,5,2).

The the distance is obtained applying the formula for the distance between two points as follows:

Distance = square root [(5 - 0)² + (5 - 5)² + (2-2)²] = 5 units.

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Answer:

c. h+j+k / 3

Step-by-step explanation:

If {v1, v2, v3} is an orthogonal basis for W, then multiplying v3 by a scalar c gives a new orthogonal basis {v1, v2, cv3 } is false statement.

Orthogonal basis:

Linear algebra, the orthogonal basis V of the inner product space is the basis of V whose vectors are mutually orthogonal.

When a vector of orthogonal basis is normalized, the resulting basis is an orthonormal basis.

let W be a vector space. The set S= {v₁,v₂,v₃} is an orthogonal basis for W.

The number of basis elements in W = 3 that is the dimensions of W is three.

If we multiply v₃ by c =0, then the set becomes

{v₁, v₂, cv₃} = {v₁, v₂,0}. The dimension of the set is 2, that is less than the dimension of W. If the scalar factor is 0, then it will not give a new orthogonal basis. Since in this case the number of basis elements will be reduced by 1 hence will change dimension of W.

Therefore, The statement is False.

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The potential energy of an rock of 40 kg and at a height of 10m is 4000 kgm^2s^2

What is potential energy and its formula?

An object possesses potential energy as a result of its position in relation to other objects. Potential energy is so named because it has the ability to transform into other types of energy, such kinetic energy. the formula for potential energy is mgh where m is the mass, g is the acceleration due to gravity and h is the height of the object.

We are given a rock of mass 40 kg which is at a height of 10m

We substitute the values in the formula to find the potential energy of the rock

Potential energy = m*g*h

Potential energy = [tex]40*10*10[/tex]

Potential energy = 4000 kgm^2s^2

Hence the potential energy of the rock is 4000 kgm^2s^2

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Answer:

below!

Step-by-step explanation:

reasons for each of the statements.

1. this is given information so there is nothing to prove here.

2. this is also given information, nothing to prove here.

3. this is also given information, nothing to prove here.

4. This is just because they are both 2 parts of the big segment (not sure what exactly is the reason your question is looking for but it something along these lines should be in your dropdown menu).

5. same as 4

6. since QR is equal to TU you can substitute TU with QR. QT stays the same.

7. Since we can see based on 5 and 6 that they have the same values by the same equations.

8. SAA

9. CPCTC

Hope this was helpful! :)

We need to prove triangles are PQU and SRT are congruent and we can see they have two angles marked as congruent and the included sides partially marked as equal and include the common segment.

We'll first prove the sides RT and QU are congruent, then prove the triangles are congruent. It will then give us the corresponding sides' congruence.

Missing reasons:

The first 3 statements reflect the given fact.

The next 2 statements reflect segment addition postulate.

After substitution of equal segments we prove that partially overlapping sides are congruent. Then it gives us the missing bit to prove the congruence of triangles.

Corresponding Parts of Congruent Triangles are Congruent.

It is possible for the the two different non-isomorphic graphs can have same number of vertices and edges , and the two such graphs are drawn below .

Two graphs are said to be isomorphic if they contain same number of vertices connected in the same way.

So, Yes the two non-isomorphic graphs can have the same number of vertices and the same number of edges.

The two such graphs are drawn below

If the degrees of vertices in two graph are same then the graph will become isomorphic it will no longer be Non-Isomorphic.

Therefore, No two graphs that are non isomorphic can have same number of edges, vertices and degrees of vertices all at once.

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The equation of the circle is [tex](x+3)^2 + (y-5)^2 = 5^2[/tex]

What does the equation for a circle look like in cartesian form?

Cartesian coordinate system with the origin indicated in red and a circle with a radius of two. A circle has the equation (x a)^2 + (y b)^2 = r^2, where a and b are the center's coordinates (a, b) and r is the radius.

The equation of a circle has the following conventional Cartesian form:

[tex](x-h)^2 + (y-k)^2 = r^2[/tex] ---------- 1

where (h, k) is the circle's center, (x, y) is any point on the circle, and r is its radius

Substitute the center (-3,5) into equation [1]:

[tex](x+3)^2 + (y-5)^2 = r^2[/tex] --------------------- 2

Substitute the point (1,8) into equation [2] and solve for r:

[tex](1+3)^2 + (8-5)^2 = r^2\\(4)^2 + (3)^2 = r^2\\16 + 9 = r^2\\25 = r^2\\r = 5[/tex]

Substitute 5 for r into equation [2]

[tex](x+3)^2 + (y-5)^2 = 5^2[/tex]

Hence the equation of the circle is [tex](x+3)^2 + (y-5)^2 = 5^2[/tex]

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Between the years 2000 and 2007, there was a total price change of 24.79.

Area is the total amount of space that an object's shape or a flat (2-D) surface occupy.

Create a square on paper by using a pencil. Tw dimensions make it up. A shape's area on paper is the space it takes up.

Imagine that your square is made up of smaller unit squares.

The area of a figure is equal to the number of unit squares required to completely cover the surface area of a particular 2-D shape. Square cms, square feet, square inches, square meters, etc. are a few common units for measuring area.

To get the area of the square figures presented below, draw unit squares with 1-centimeter sides. Therefore, the shape will be measured.

Hence, Between the years 2000 and 2007, there was a total price change of 24.79.

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Answer:

Step-by-step explanation:

The graph of the function g(x)=(x-5)²+1 is the graph of the function f(x)=x², shifted 5 units to the right along the x-axis and shifted up 1 unit along      the y-axis

To prove that the triangles are congruent by ASA, the statement and reason could be used as part of the proof is option (A) ∠FGH ≅ ∠KGJ because vertical angles are congruent.

Here we have two triangle FHG and KJG,

The sides HF and the side JK are parallel

The length of the side of the HG and the length of the side JG are equal

HG ≅ JG

According to the ASA property of congruent, if the two angle and one side of the triangle are equal to the two angle and one side of the other triangle, then the both triangles are congruent

Here we have to prove that angle FHG and angle KJG are vertical angles

Here the vertical angles are congruent

Therefore, the statement that can be the part of proof is option (a) angle FGH ≅ angle KGJ because vertical angles are congruent.

I have solved the question in general, as the given question is incomplete

The complete question is:

Given: HF || JK; HG ≅ JG

Prove: FHG ≅ KJG

To prove that the triangles are congruent by ASA, which statement and reason could be used as part of the proof?

a) FGH ≅ KGJ because vertical angles are congruent.

b) JKG ≅ HFG because vertical angles are congruent.

c) FHG ≅ JKG because right angles are congruent.

d) HFG ≅ KJG because alternate interior angles are congruent.

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