What are special segments of a triangle?

On observing the provided question we can say that - the function f(x) = [tex]a^x[/tex] is Domain: (−∞,∞) or all real numbers

Range: the discrepancy between the top and bottom numbers. To get the range, locate the greatest observed value of the variable and deduct the least observed value (the minimum). The data points between the two extremes of the distribution are not taken into consideration by the range; just these two values are considered. Between the lowest and greatest numbers, there is a range. Values at the extremes make up the range. The data set 4, 6, 10, 15, 18, for instance, has a range of 18-4 = 14, a maximum of 18, a minimum of 4, and a minimum of 4.

The answer to the question is that all real numbers or (−∞,∞)

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The common multiple of 5y²- 45 and y + 3 is (y + 3).

For any two or more expressions, if any one of their factors is common in all of them, then that factor is said to be the common multiple of those expressions.

The given expressions are 5y²- 45 and y + 3

On factorizing them, we get

5y²- 45 = 5(y² - 9)

⇒ 5(y² - 3²)

From he binomial formula a² - b² = (a + b)(a - b), we can write

5y²- 45 = 5(y + 3)(y - 3)

Now, the second expression has no further simplification. The expression is itself a factor for it.

So, we can write their factors as

5y²- 45 = 5 × (y + 3) × (y - 3)

y - 3 = 1 × (y - 3)

From this, (y - 3) is the only factor that is a common factor for both the expression.

So, (y - 3) is the common multiple for the given expressions.

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By drawing a mirror line across the center of a shape and examining if both halves are the same, you may determine whether it is symmetrical.

When anything has two identical parts, it is symmetrical. In other terms, symmetry is the presence of identical pieces facing one another or revolving around an axis.

But what attributes give a shape symmetry? A symmetrical shape has an identical side on the side, also known as symmetry. When mirrored, rotated, or translated, symmetrical shapes retain their original appearance. The four main subcategories of symmetry are translation, rotation, reflection, and glide reflection. However, the primary type of symmetry in mathematics taught in schools is reflectional symmetry, commonly referred to as mirror symmetry or line symmetry.

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The point lies in the solution set if x represents the number of children's books and y represents the number of puzzle books Amelie buys (9,5).

To find the point that lies in the solution set, you need to consider the constraints given in the problem. Amelie wants to purchase at least 2 more children's books than puzzle books, which means that the number of children's books should be greater than or equal to the number of puzzle books plus 2. She can afford no more than 15 items total, which means that the total number of children's books and puzzle books should be less than or equal to 15.

Based on these constraints, the point that lies in the solution set is (9,5). This point satisfies both constraints, because the number of children's books (9) is greater than the number of puzzle books (5) plus 2, and the total number of children's books and puzzle books (14) is less than or equal to 15. Therefore, the correct answer is (d) (9,5).

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Answer:

25

Step-by-step explanation:

(60+15)/3-2(4)+2^3

75/3 -2 x4 + 2^3

75/3 - 8 +2^3

75/3 - 8 +8

75/3

25

Step-by-step explanation:

To multiply 17m 25cm by 24, you need to first convert the units of measurement to a common unit, such as meters or centimeters.

17m 25cm can be written as 17.25 m, since 1 meter is equal to 100 centimeters. Therefore, to multiply 17.25 m by 24, you can simply perform the calculation as follows:

17.25 m * 24 = 413 m

This means that the result of the multiplication is 413 meters.

Alternatively, you could express the result in centimeters by performing the calculation as follows:

(17 m * 100 cm/m) + (25 cm * 24) = 1700 cm + 600 cm = 2300 cm

This means that the result of the multiplication is 2300 centimeters.

The surface area of the net of the triangular prism in the space provided,

2400cm².

Prism is a polyhedron with two identical ends in three dimensions.

Surface area = (Perimeter of the base × Length of the prism) + (2 × Base                Area)

                     = (S₁ + S₂+ S₃)l + bh

where, b is the bottom edge of the base triangle, h is the height of the base triangle, l is the length of the prism and S₁, S₂, and S₃ are the three edges (sides) of the base triangle (bh) is the combined area of the two triangular faces [2 × (1/2 × bh)] = bh.

The side of the triangle is S₁ = 8, S₂= 15, and S₃= 17.

The rectangular's base = 20 and height = 8

Substituting all the values in the formula:

Surface area

= (Perimeter of the base × Length of the prism) + (2 × Base Area)

= [ ( 15 +17+8) x 20] + (20 x 80)

= 800 + 1600

= 2400 cm²

Therefore, the surface area of the prism is 2400 cm².

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x=-5/√2 and x=-√2 are roots of the quadratic equation √2x²+7x+5√2=0

A quadratic equation is a second-order polynomial equation in a single variable x , ax2+bx+c=0. with a ≠ 0 .

The given quadratic equation is √2x²+7x+5√2=0

We will factorize  the middle term by middle term method

√2x²+2x+5x+5√2=0

√2x(x+√2)+5(x+√2)=0

(√2x+5)(x+√2)=0

√2x+5=0

√2x=-5

Divide both sides by √2

x=-5/√2

and x+√2=0

x=-√2

Hence, x=-5/√2 and x=-√2 are roots of the quadratic equation √2x²+7x+5√2=0

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Answer: 38

Step-by-step explanation:

24 divided by 5 = 4.8 per hour

4.8 times 8 = 38.4 or 38 in 8 hours

It can burn 38 in 8 hours

To round to the nearest meter, use the units place and round to the nearest whole number. At the moment, we have a two in place of a one.

It is two divisions or two tenths of a mile short of the halfway point. Therefore, seven centimeters are the closest centimeters.

To figure out how to round, we look to the right, at the unit in the tens place.

Let x represent the unknown, which is the building's height.

level of post = level of the structure

level of the shadow level of the shadow

3.3 m = x

1.78 m 49. 75 m Find the solution for x: (3.3 m) (49.75 m) = x 1.78 m = x The building is 92 meters high, rounded to the nearest meter.

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38 °F is the temperature after 5 hours of time.

Two or more expressions with an Equal sign is called as Equation.

Given that the temperature is 36 °F.

The temperature decreases for 4 hrs

Each hour, the temperature decreases by 2 °F.

1 hour=  2 °F.

For 4 hours it will be 4(2)=8 °F.

So the temperature after 4 hours is 36-8=28 °F.

the temperature rises 10 °F in one hour

28 °F+10°F=38 °F

Hence, the temperature after 5 hours is 38 °F.

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Triangle ABC is congruent to Triangle CDA using the SSS rule.

It is a two-dimensional figure which has three sides and the sum of the three angles is equal to 180 degrees.

We have,

SSS rule states that,

If the three sides of one triangle are equal to the three sides of another triangle, then the triangles are congruent.

To prove:

ΔABC ≅ ΔCDA.

AB = DC ( given)

AD = BC (given)

AC = AC (common to both the triangles)

ΔABC ≅ ΔCDA ( By SSS rule )

Thus,

ΔABC ≅ ΔCDA by SSS rule.

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Estimate the height of the tree 50 years after it was planted is 15.92 feet

DBH (inches) × π = Estimated Age of Tree (years)

Let us Consider inches is X

X *3.14=50

X=50/3.14

X=15.92

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The first step in solving a polynomial inequality is to determine the type of inequality that you are dealing with. There are two types of polynomial inequalities: strict inequalities and non-strict inequalities.

Strict inequalities are inequalities that are either less than or greater than a value, but not equal to it. For example, the inequality x < 4 is a strict inequality because it means that x is strictly less than 4 (i.e., it is less than 4 but not equal to 4).

Non-strict inequalities are inequalities that are either less than or equal to, or greater than or equal to a value. For example, the inequality x <= 4 is a non-strict inequality because it means that x is less than or equal to 4 (i.e., it is either less than 4 or equal to 4).

Once you have determined the type of inequality that you are dealing with, you can proceed to the next step, which is to solve the inequality by isolating the variable and performing the necessary operations to find the solution.

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The polynomial p(x) is x³ - 7x - 6 that is (x+1) (x+2)(x - 3)

An expression that consists of variables, constants, and exponents that is combined using mathematical operations like addition, subtraction, multiplication, and division is referred to as a polynomial (No division operation by a variable).

Given that polynomial:

p(x) = x³ - 7x - 6

Known factor: (x+1)

polynomial equation expressed as the sum of linear components.

The degree of polynomial is 3, and when we divide it by a linear equation, the result will be a quadratic.  That quadratic will have 2 solutions.

We can solve the quadratic in two linear factors and as a result we will have the answer.

First of all, we need to divide the polynomial  p(x)with the given factor (x+1) to find the other factors.

x³ - 7x - 6/x+1 = x² - x -6

Solving the quadratic:

x²- x - 6 = x² -3x +2x - 6

x(x-3)+2(x-3)

(x+2)(x-3)

Therefore the polynomial will be p(x) = x³ - 7x - 6 = (x+1) (x+2)(x - 3)

The complete question is : The polynomial p ( x ) = x 3 − 7 x − 6 p(x)=x 3 −7x−6p, left parenthesis, x, right parenthesis, equals, x, cubed, minus, 7, x, minus, 6 has a known factor of ( x + 1 ) (x+1)left parenthesis, x, plus, 1, right parenthesis. Rewrite p ( x ) p(x)p, left parenthesis, x, right parenthesis as a product of linear factors. p ( x ) = p(x)=p, left parenthesis, x, right parenthesis, equals

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Therefore , the kids can able to sings alphabets ABC by the age of 2 and by age of 4 children frequently know every letter of the alphabet .

In alphabet, set of graphs, or characters, used to represent the phonemic structure of a language .The characters are often placed in a specific order, or sequence, in most alphabets (e.g., A, B, C, etc.).

Here,

By the age of two, children can sing or say out loud the "ABC" song and begin to recognise some letters.

By the age of three, children may recognise around half of the alphabet's letters and begin to associate letters with their sounds.

(S makes the sound /s/.)

Most kids are familiar with all 26 letters of the alphabet and their order by the time they are four years old.

Therefore , the kids can able to sings alphabets ABC by the age of 2 and by age of 4 children frequently know every letter of the alphabet .

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Children can begin to sing the alphabet by the time they are two years old.

The phonemic structure of a language is represented by an alphabet, which is a system of graphs or symbols. In most alphabets, the characters are frequently arranged in a certain order, or sequence (e.g., A, B, C, etc.).

There are 26 alphabets named from A to Z.

By the time they turn two, kids can sing or say the "ABC" song aloud and start to recognize some letters.

Around half of the alphabet's letters may be recognized by three-year-old, and they are starting to connect letters with sounds.

By the time they are four years old, the majority of children are familiar with all 26 letters of the alphabet and their placement.

As a result, children can begin to sing the alphabet by the time they are two years old, and by the time they are four, many of them are already familiar with all 26 letters.

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The Answers to the Equation is shown below.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides. LHS = RHS is a common mathematical formula.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given:

1. 5x - 7 = 4x +3

5x- 4x = 3 + 7

x= 10

2. 6 + 2x = 7x - 9

7x- 2x = 6+9

5x = 15

x= 3

3. 8x + 1 = - 8 - x

9x = -9

x= -1

4. -5+ 12x = 18x + 7

6x = -12

x= -2.

5. -4x+ 3= 5x- 13 -x

-8x = -16

x= 2

6. -10 - 11x + 24 = 3x

14x = 14

x=1

7. 8 + 3x = x + 11 + 2x

No Solution

8. 5 +17x +9 = 2x + 21x + 14

6x= 0

x=0

9. 6x + 8 - 13x + 10 = 4 - 6x - 11 + 4x

-5x= 25

x= -5

10. -14 + 5 - x - 3 =x+6+3x + x

4x= -14

x= -3.5

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Answer: a) The equation of the least-squares regression line is y = -1.66x + 78.83. b) Residual =  7.11 inches. c) The residual is the deviation of a data point from the fitted regression line. In this case, the residual is 7.11 inches, indicating that the player who wears a size 9 and is 71 inches tall is taller than what would be expected based on the trend in the data.

Step-by-step explanation:

What is the least square regression and residual?

Least squares regression is a statistical method that is used to find the line of best fit for a set of data points. It involves finding the line that minimizes the sum of the squares of the differences between the actual data points and the predicted values on the line. The residual is the difference between the actual data point and the predicted value on the line.

Part A: The equation for the least-squares regression line can be found by using the following formula: y = bx + a

To find the values for a and b, we need to calculate the following values:

x: the mean of the shoe sizes,

y: the mean of the heights.

S x y: the sum of the products of the differences between the shoe sizes and x and the differences between the heights and y.

S x x: the sum of the squares of the differences between the shoe sizes and x.

X = (7+10+12+8+9.5+10+11.5+12.5+13.5+10) / 10 = 10.5

Y = (64+69+72+67+69+70+75+73+71+68) / 10 = 69.4

To find S x y and S x x, we use:

S x y = ∑(x - x)(y - y)

S x x = ∑(x - x)^2

Inserting the values from the table, we get:

S x y =-47.5

S x x = 28.5

Using values, we can calculate the slope and y-intercept of the least-squares regression line:

b = S x y / S x x = -47.5 / 28.5 = -1.66

a = y - bx = 69.4 - (-1.66)(10.5) = 78.83

So the equation of the least-squares regression line is y = -1.66x + 78.83

Part B: We can use the equation for the least-squares regression line to determine the residual for a player who wears a size 9 and is 71 inches tall. We calculate the player's height using the size 9 into the equation:

y=-1.66x+78.83=63.89 inches.

The residual is then calculated by deducting the expected height from the actual height: 7.11 inches =71- 63.89 inches.

Part C: The residual is the variation between the dependent variable's actual value (height) and its anticipated value according to the least-squares regression line. The athlete in this instance, who wears a size 9 and is 71 inches tall, is taller than what might be anticipated based on the trend in the data, according to the residual of 7.11 inches.

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The equation of the least-squares regression line is y = -1.66x + 78.83. b) Residual =  7.11 inches.

What is the least square regression and residual?
Least squares regression is a statistical method that is used to find the line of best fit for a set of data points. It involves finding the line that minimizes the sum of the squares of the differences between the actual data points and the predicted values on the line. The residual is the difference between the actual data point and the predicted value on the line.

Part A: The equation for the least-squares regression line can be found by using the following formula: y = bx + a
To find the values for a and b, we need to calculate the following values:
x: the mean of the shoe sizes,
y: the mean of the heights.
S x y: the sum of the products of the differences between the shoe sizes and x and the differences between the heights and y.
S x x: the sum of the squares of the differences between the shoe sizes and x.
X = (7+10+12+8+9.5+10+11.5+12.5+13.5+10) / 10 = 10.5
Y = (64+69+72+67+69+70+75+73+71+68) / 10 = 69.4
To find S x y and S x x, we use:
S x y = ∑(x - x)(y - y)
S x x = ∑(x - x)^2
Inserting the values from the table, we get:
S x y =-47.5
S x x = 28.5
Using values, we can calculate the slope and y-intercept of the least-squares regression line:
b = S x y / S x x = -47.5 / 28.5 = -1.66
a = y - bx = 69.4 - (-1.66)(10.5) = 78.83
So the equation of the least-squares regression line is y = -1.66x + 78.83
Part B: We can use the equation for the least-squares regression line to determine the residual for a player who wears a size 9 and is 71 inches tall. We calculate the player's height using the size 9 into the equation:
y=-1.66x+78.83=63.89 inches.
The residual is then calculated by deducting the expected height from the actual height: 7.11 inches =71- 63.89 inches.
Part C: The residual is the variation between the dependent variable's actual value (height) and its anticipated value according to the least-squares regression line. The athlete in this instance, who wears a size 9 and is 71 inches tall, is taller than what might be anticipated based on the trend in the data, according to the residual of 7.11 inches.

c) The residual is the deviation of a data point from the fitted regression line. In this case, the residual is 7.11 inches, indicating that the player who wears a size 9 and is 71 inches tall is taller than what would be expected based on the trend in the data.

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The HCF and LCM are Highest common factor and Least common multiple respectively

What is HCF and LCM?

The Greatest Common Divisor GCF or the Highest Common Factor HCF is the highest number that divides exactly into two or more numbers. It is also expressed as GCF or HCF

Least Common Multiple (LCM) is a method to find the smallest common multiple between any two or more numbers. A common multiple is a number which is a multiple of two or more numbers

Product of HCF x LCM = product of two numbers

Given data ,

Let the two numbers be A and B

The value of A is A = 160

The value of B = 252

HCF is the product of common prime factors

So , Prime Factorization of 160: 2 x 2 x 2 x 2 x 2 x 5

And , Prime Factorization of 252: 2 x 2 x 3 x 3 x 7

So , HCF = 2 x 2 = 4

Now, finding LCM of 160 and 252

LCM is the product of all factors of both numbers

Prime Factorization of 160: 2 x 2 x 2 x 2 x 2 x 5

Prime Factorization of 252: 2 x 2 x 3 x 3 x 7

LCM= 2 x 2 x 2 x 2 x 2 x 3 x 5 x 7

LCM= 10080

Now , the product of HCF x LCM = product of two numbers

160 x 252 = 40320

4 x 10080 = 40320

Hence , the HCF and LCM are solved

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Factoring x³ is a relatively straightforward process.

Factory is a quantity or an expression that is multiplied to another quantity or expression to obtain a product. Factor are used in mathematics to solve problems involving multiplication division and equation factor can also be used in financial analysis to calculate interest, payments loan repayments, and another financial transaction factor can also be used to analyze supply and demand market trends and other economic indicators.

The most common way to factor x³ is to use the factorization method. This involves writing x³ as a product of three linear factors, which can be done in the following way:

x³ = x × x × x

Alternatively, you can use the method of grouping to factor x³. This involves writing x³ as a product of two binomials, which can be done in the following way:

x³ = (x × x) × x

You can also use the method of synthetic division to factor x³. This involves dividing the coefficient of x³ by the coefficient of x and then dividing the resulting coefficients by the coefficient of x. This can be done in the following way:

x³ = x² + 0 × x + 0

Once you have factored x³, you can use the factors to solve equations or to find other information related to the equation. In addition, you can use the factored form of x³ to simplify certain expressions.
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A discontinuity in mathematics is when a function is not well defined

As earlier defined, discontinuity deals with a function that is not well defined.  

Different functions have different discontinuity depending on the function

Good examples of discontinuity include

y=1/x this is a discontinuous function.

x=0 is also a discontinuous function as it is not well defined. because there is no value for y

The function TanA is a discontinuous function because it is not well defined.

Other discontinuities include

y=5п/3,  9п/4, 6п8, 4п/3

All these are discontinuous functions

In conclusion, a  discontinuity in mathematics is when a function is not well defined

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For the shaded region given in the diagram the area is obtained as 50.2 cm². The correct option is (A).

A circle is a geometric shape, all of which points are equidistant from a fixed point called as the center.The center of the circle does not lie on it. The angle subtended by the semicircle on its any point is always a right angle.

The diameters of the external and internal circle are 10 cm and 6 cm.

Then, the radius are given as 10/2 = 5 cm and 6/2 = 3 cm

Now, area of the shaded region is given as follows,

Area of external circle - Area of internal circle

= π × 5² -  π × 3²

= 16π

= 50.2

Hence, the area of the shaded region is given as 50.2 cm².

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The level of a line's slope indicates its steepness.

The slope is computed mathematically as "rise over run" (change in y divided by change in x).

In accordance with the slope equation, the slope of such a line may be determined by dividing the rise of the line between two positions by the run of the same line between the same two sites. 

That is to say,

Example -

This line goes through the points (0,0) and (4,4).

Now, slope = (y₂ - y₁) / (x₂ - x₁)

slope = (4 - 0) / (4 - 0)

slope = 4 / 4

slope = 1

So, from the above calculation it is shown that, the slope of the line is 1.

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For a given function f, it is linear if the function has a constant slope and it doesnot contain the power of x greater than 1.

An association between each element of a non-empty set A and at least one element of a different non-empty set B is called a function. The core concept of mathematics' calculus is functions. The unique varieties of relations are the functions. In mathematics, a function is represented as a rule that produces a distinct result for each input x.

A function which is also linear, represents a straight line in the graph. The graphs to the linear functions are straight lines which have constant slope throughout the graph.

A function f is referred to as linear if it meets both of the following criteria:

The slope of the function is constant. This indicates that the slope of the line is constant, independent of the value of x that you enter into the function.

There are no powers of x bigger than 1 in the function. This indicates that there are no terms in the function that have x2, x3, etc.

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First make the graph of the functions, then  if the two graphs are symmetric with respect to the line y = x (mirror images over y = x ), then they are inverse functions.

Symmetric

A figure or shape that can be divided into two equal parts by a line is called symmetric figures.

Inverse Functions

The inverse function of a function f  is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by F^-1.

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Answer:

8cm

Step-by-step explanation:

We can figure out what the side length of this would be by squaring the area.

[tex]\sqrt{64}[/tex] = 8

To check our work, we can multiply the length by the height. (8 and 8)

8 × 8 = 64

Nado's chessboard side length would be 8cm.

The area of segment enclosed by chord [BC] is 0.94 [tex]cm^{2}[/tex]

Segment and area of a circle segment:

A segment is essentially the area of a circle between the chord and the arc. A circle is a path that a point equidistant from a single point on the plane can follow; this point is known as the circle's center, and the distance between it and that point is known as the circle's radius. A segment is a section of a circle's interior. A sector is the region that a segment encloses and the angle that a segment occupies. By deducting the triangle formed inside the sector from the sector where the segment is located, one can calculate the area of a circular segment.

Given radius and angle subtended by arc on circle is 4 cm and 55° respectively.

So the length of chord [BC] is [tex]\frac{55}{180}[/tex]×π = 3.84 cm.

Required area of segment enclosed = Area of segment subtended by arc [BC] - area of ΔABC

>semi-perimeter of ΔABC is [tex]\frac{(4+4+3.84)}{2} = 5.92[/tex]

>Area of ΔABC is [tex]\sqrt{5.92*(5.92-4)*(5.92-4)*(5.92-3.84)} = 6.74 cm^{2}[/tex]

>Area of segment subtended by arc {BC} = θ/360° × π[tex]r^{2}[/tex]

                                                                     =[tex]\frac{55}{180}[/tex]×16π

                                                                     = 7.68[tex]cm^{2}[/tex]

∴The area of segment enclosed by chord [BC] is [tex]7.68-6.74 = 0.94 cm^{2}[/tex]

The area of segment enclosed by chord [BC] is 0.94 [tex]cm^{2}[/tex]

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