If the points (2, 5), (1, 1) and (a, 3) are collinear, find the value of a.​

The slope of the original line is 3/5.

What is the slope?

The slope of a line is a measure of its steepness. Mathematically, the slope is calculated as "rise over run" (change in y divided by change in x).

The slope of a line perpendicular to another line can be found by taking the negative reciprocal of the slope of the original line.

To find the slope of the original line, we need to convert the equation to the slope-intercept form (y = mx + b), where m is the slope.

To convert, we'll isolate y:

-3x + 5y = 20

5y = 3x + 20

y = (3/5)x + 4

The slope of the original line is 3/5. The slope of the line perpendicular to this line would be the negative reciprocal of this slope, or -5/3.

Hence, The slope of the original line is 3/5.

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The differential equation for the time evolution of the toxic chemical in the pond is dy/dt = -200y + (100/2000)x

where y represents the amount (in kg) of the contaminant in the pond and x represents the amount of contaminant added to the pond per minute (5 kg/minute in this case). The negative sign in the equation indicates that the amount of contaminant is decreasing in the pond, while the positive term (100/2000)x indicates that the contaminant is being added to the pond at the rate of 5 kg/minute.

To solve for y, we can integrate the equation with respect to time. We start by multiplying both sides by dt, giving us the integral form of the equation: dy = -200y dt + (100/2000)x dt. Integrating both sides gives us the solution y = -200y(t) + (100/2000)x(t) + C, where C is an integration constant. Since we know the initial amount of contaminant in the pond is 5 kg, we can set y(0) = 5 and solve for C, giving us C = 5. Therefore, the solution to the differential equation is y(t) = -200y(t) + (100/2000)x(t) + 5.

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Answer: 3

Step-by-step explanation:

[tex]f(0)=3(0)-3=-3\\\\\implies g(f(0))=g(-3)=-(-3)=3[/tex]

The statement, "two triangles with corresponding congruent angles are congruent," is true.

According to the Triangle Congruence Postulate, if two triangles have three pairs of corresponding congruent angles, then the triangles are congruent.

This means that if two triangles have three pairs of corresponding congruent angles, all six angles are congruent and the triangles are exactly the same size and shape.

Thus, the statement, "two triangles with corresponding congruent angles are congruent," is true.

Congruence is determined by the congruence of angles, sides, and the overall shape of the triangle. If two triangles have the same angles and sides, then they are considered to be congruent.

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1) The function f(x) is defined as follows: B. F(x) = -(x + 2)²(x - 2).

2) The standard definition of function f(x) is given as follows: A. F(x) = ax³ + bx² + cx + d.

4) The approximate solution to the system of equations is given as follows: A. (-3.43, 6.30).

The function f(x) is defined according to the Factor Theorem, with the product of the linear factors of the function, which are dependent on the roots of the functions.

The zeros of the function, along with their multiplicity, are given as follows:

Hence the function is defined as follows:

F(x) = a(x + 2)²(x - 2).

The y-intercept is positive, meaning that:

-8a > 0.

Thus the leading coefficient a is negative, and the correct option is given by option B.

A cubic function, in which the coefficients a and d are positive, is represented as follows:

A. F(x) = ax³ + bx² + cx + d.

The solution to the system of equations is found through graphing, and is the point of intersection of the two functions.

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12. i) By walking the 20 kilometers in the charity walk, Vreni raised $250 at a unit rate of $12.50 per kilometer.

12. ii) The total money raised by all participants in the charity walk was $3,100, using proportions.

12. iii) The percentage increase on the previous year's total is 26.53%.

Part B:

11. i) Based on a ratio of 3:2 of road and footpath distance, the road distance covered 12 kilometers.

11. ii) Using the distance, time, and speed formula, Vreni took 7 hours and 12 minutes to walk the 20 kilometers.

11. iii) Vreni's average speed is 2.78 km/h.

11. iv) Vreni finished the charity walk at 16:07.

Part C: The scale of the map to the total distance covered by Vreni was 1: 4 or a scale factor of 25%.

The ratio is the quotient of a value compared to another.

Ratios are proportionate values, which are equated to each other.

The average speed and scale factor show the ratios of one value and the whole.

12) Charity Walk and Vreni:

The total distance walked = 20 kilometers

The unit rate of raising money per kilometer = $12.50

i) The total amount raised = $250 ($12.50 x 20)

Money raised in 12. 1 = 5/52 of the total money raised

ii) The total money raised in the charity walk = $3,100 ($250 ÷ 5/52)

iii) The total amount raised in the previous year = $2,450

The total amount raised in the current year = $3,100

Difference between current and previous year = $650 ($3,100 - $2,450)

Percentage increase on the previous year's total = 26.53% ($650/$2,450 x 100)

11b) Ratio of road and footpath walk = 3:2

The sum of ratios = 5

Road distance = 3/5 x 20 = 12 kilometers

Footpath distance = 2/5 x 20 = 8 kilometers

Vreni's walking speed along the road = 3 km/h

Road distance = 12 kilometers

Time to walk the road = 4 hours (12/3)

Vreni's walking speed along the footpath = 2.5 km/h

Footpath distance = 8 kilometers

Time to walk the footpath = 3.2 hours or 3 hours and 12 minutes (8/2.5) or 192 minutes (8/2.5 x 60)

Total time spent by Vreni = 7 hours and 12 minutes (4 hr + 3 hr + 12 min)

Vreni's average speed = Distance/Time

= 20 km/7.2 hrs

= 2.78 km/h

Starting time = 08:55

Total hours = 7 hours and 12 minutes

Finishing time = 16:07 (08:55 + 7 hrs + 12 min)

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Answer:

-15

Step-by-step explanation:

im thinking +[-15] and [-15] are the same thing

Answer:

If two triangles are similar, the ratio of their corresponding side lengths is the same as the ratio of their areas.

We know that the area of the first triangle is 18 cm^2, and the area of the second triangle is 8 cm^2. So, the ratio of their areas is 18/8 = 9/4

We also know that one of the sides of the first triangle is 4.5 cm.

So the ratio of the corresponding side of the second triangle to the first triangle is:

(length of corresponding side of second triangle) / 4.5 = 9/4

We can cross-multiply and solve for the length of the corresponding side of the second triangle:

4.5 * (9/4) = 9 * (9/4) / 4 = 9 * 2.25 / 4 = 20.25 / 4 = 5.0625 cm

So, the length of the corresponding side of the second triangle is 5.0625 cm.

As, the moon has an elliptical orbit around a planet, the maximum distance between the planet and its moon is 88.54 Mm.

What is an elliptical orbit?

When an object moves around another object in an oval shaped path (ellipse), it is known to be revolving in an elliptical orbit.

The standard form of the equation of an ellipse with center at the origin (0,0) and x - axis major axis is given by

x²/a² + y²/b² = 1

Now,

Given that a moon's elliptical orbit around a planet is modeled by the equation 225x² + 576y² = 176,400, where distance is measured in megameters (Mm). and the planet is the center of the orbit's path.

Since the equation of the path is the equation of an ellipse, we write it in standard form

As standard form of the equation of an ellipse with center at the origin (0,0) and x - axis major axis is

x²/a² + y²/b² = 1  (1) where

a = vertex of major axis and

b = vertex of minor axis

So, converting 225x² + 576y² = 176,400 into standard form, we have

225x² + 576y² = 176,400

225x²/1764,000 + 576y²/1764,000 = 176,400/1764,000

x²/7840 + y²/3062.5 = 1  (2)

Comparing equation (2) with (1), we have that

a² = 7840 and b² = 3062.5

⇒ a = √7840 = 88.54 and

⇒ b = √3062.5 = 55.34

Since a = 88.54 is greater than b = 55.34, so, a is the major axis, and the maximum distance between the planet and its moon is 88.54 Mm

So, the maximum distance between the planet and its moon is 88.54 Mm

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Since the moon's has an elliptical orbit around a planet, the maximum distance between the planet and its moon is 88.54 Mm

How to find the maximum distance between the planet and its moon?

x²/a² + y²/b² = 1  (1) where

a = vertex of major axis and

b = vertex of minor axis

o, converting 225x² + 576y² = 176,400 into standard form, we have

225x² + 576y² = 176,400

225x²/1764,000 + 576y²/1764,000 = 176,400/1764,000

x²/7840 + y²/3062.5 = 1  (2)

Comparing equation (2) with (1), we have that

a² = 7840 and b² = 3062.5

⇒ a = √7840 = 88.54 and

⇒ b = √3062.5 = 55.34

Since a = 88.54 is greater than b = 55.34, so, a is the major axis, and the maximum distance between the planet and its moon is 88.54 Mm

So, the maximum distance between the planet and its moon is 88.54 Mm

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The value of x using trigonometry is √6.

The study of correlations between triangles' side lengths and angles is known as trigonometry. The field was created in the Hellenistic era in the third century BC as a result of the use of geometry in astronomical research.

The area of mathematics known as trigonometry examines the link between the ratios of a right-angled triangle's sides to its angles. Trigonometric ratios, such as sine, cosine, tangent, cotangent, secant, and cosecant, are employed to analyze this connection.

The measurement of angles and issues relating to angles are covered in the fundamentals of trigonometry. Trigonometry has three fundamental operations: sine, cosine, and tangent. The cotangent, secant, and cosecant are three crucial trigonometric functions that may be derived from these three fundamental ratios or functions. These functions serve as the foundation for all the key ideas in trigonometry.

In the triangle base = [tex]\sqrt{2}[/tex] and height as x

The value of the angle is [tex]60\x^{o}[/tex]

so tanθ = tan[tex]60\x^{o}[/tex]

[tex]\frac{height }{base} = tan60\x^{o} \\[/tex]

⇒ x = √3 * √2

⇒ x = √6

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Answer:

simple 5×-5 the answer divide by -5

The null hypothesis is that the population mean is equal to 3 minutes. The calculated p-value from the sample data is 0.28, which is higher than the 0.05 significance level.

not significantly more than 3 minutes.

The null hypothesis is that the population mean is equal to 3 minutes. The calculated p-value from the sample data is 0.28, which is higher than the 0.05 significance level. Therefore, we cannot reject the null hypothesis and conclude that the mean of the population is not significantly more than 3 minutes.

1. The null hypothesis is that the population mean is equal to 3 minutes.

2. A sample of 100 customers was taken, with an average length of calling time of 3.1 minutes and a sample standard deviation of 0.5 minutes.

3. The p-value from the sample data is 0.28, which is higher than the 0.05 significance level.

4. Therefore, we cannot reject the null hypothesis and conclude that the mean of the population is not significantly more than 3 minutes.

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Answer:

whats ..... representating

is it a continuous series?

Answer:

(- 1, 5 )

Step-by-step explanation:

y = x + 6 → (1)

y = 2x + 7 → (2)

since both equations have y on the left side, then equate the right sides

2x + 7 = x + 6 ( subtract x from both sides )

x + 7 = 6 ( subtract 7 from both sides )

x = - 1

substitute x = - 1 into either of the 2 equations for y

substituting into (1)

y = x + 6 = - 1 + 6 = 5

solution is (- 1, 5 )

Step-by-step explanation:

We would make these two equations equal to each other since both equations are in y= form

[tex](x + 6) = (2x + 7)[/tex]

Now to get numbers on one side and X's on the other.

Let's start by subtracting 6 from both sides

[tex]x = 2x + 1[/tex]

Next we can subtract 2X

[tex] - x = 1[/tex]

Last but not least, we can divide by -1 to reverse the signs

[tex]x = - 1[/tex]

Answer:

80% of the flour is left.

Answer:

I hope may answer is correct! I am sorry if it is not!

The chef used 800g out of a 5kg bag of flour, so the amount of flour left is 5,000g - 800g = 4,200g. To find the percentage of flour left, divide the amount of flour left by the total amount of flour and multiply by 100: (4,200g / 5,000g) * 100 = 84%. So 84% of the flour is left.

The rate of the plane in still air is 1,000 kph and the speed of the wind is 250 kph. The result is obtained by using elimination and substituting method.

To eliminate equations, we can either add or subtract the equations. Substitution is substituting the value of one variable in the other equation.

A plane can travel:

Find the rate of the plane in still air and the speed of the wind!

Let's say

The speed of the plane with the wind is

v + w = 3750 km/3 h

v + w = 1250 kph ... (equation 1)

The speed of the plane against the wind is

v - w = 3750 km/5 h

v - w = 750 kph ... (equation 2)

Eliminate the equation 1 and 2!

v + w = 1250

v - w = 750 +

2v = 2000

v = 1000 m/s

Substitute the value of v in equation 1!

v + w = 1250

1000 + w = 1250

w = 1250 - 1000

w = 250 m/s

Hence, the speed of the plane when there is no wind and the speed of the wind respectively are 1000 m/s and 250 m/s.

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Answer: A rectangle has two dimensions: length and width. The area of a rectangle is given by the formula A = L * W, where L is the length and W is the width.

Given the area can be represented by the expression 16x^4 + 48x^3 + 82, we can see that it's a polynomial of 4 degree.

So, one of the dimensions (L or W) must be represented by a polynomial of degree 4 and the other dimension by a polynomial of degree 3.

So the expressions that could represent one of the dimensions of the rectangle are:

L = 16x^4

W = 48x^3 + 82

It's important to note that this is just one of the possible representations of the dimensions of the rectangle, depending on the values of x.

Step-by-step explanation:

Answer:

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

The scale factor is 2, since each side is twice the initial length.

There are 4 congruent copies, as we can see on the picture.

The dilated area is 12*4 = 48 square units, since the area is the product of two dimensions, each dimension is twice the initial value therefore the area is 2*2 = 4 times greater.

The distance travelled is 1035 meters, The period T is 1.67m/s and The wavelength is 0.575 meters.

Because sound is a wave, we can use the proper definitions of the terms to derive the relationship between the wavelength, frequency, and speed of a wave. The wavelength is the distance travelled by the wave in one time period of its cycle, whereas the frequency is the number of cycles per second.

Formula used:

Speed = Distance/time

Frequency of wave = 1/Time period of the wave

Wavelength = Distance travelled by the wave in one time period

Given,

a)The distance travelled is 2d,

where d is the distance to the wall

Thus d = 345 × 6/2= 1035 m

b)The frequency is 600 Hz thus the period T = 1/f= 1.67ms

c)The wavelength λ = Velocity/Frequency

= 345/600

= 0.575 meters

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Therefore , r its derivation are not present. The dependent variable does not have a transcendental function.

Numbers and their variants, math and nearby tissues, shapes and their real positions, as well as potential placements, are all studied in mathematics. The term "function" describes the connection between a group of inputs, each of which has a corresponding output. An input-output relationship is called a function when each input results in a single, unique output. A realm and a city or municipality, or scope, are assigned to each function.

Here,

Given :  (1+y²)(d²y/dt²)+t(dy/dt)+y=et

The second derivative is the highest derivative in this fractional derivative (). As a result, this differential equation has an order of 2.

The dependent variable's product and its derivative are present, indicating linearity.

The differential equation is hence non-linear.

Second order nonlinear ordinary differential equations are categorized as such.

t²(d²y/dt²)+t(dy/dt) + 2y =sint

The differential equation shown is  t²(d²y/dt²)+t(dy/dt) +2y=sint

Order: This differential equation has a higher order differential equation since the largest derivative it has is.

Lack of the dependent variable's product and/or derivative indicates linearity. Higher powers of the variable or its derivation are not present. The dependent variable does not have a transcendental function.

Therefore , r its derivation are not present. The dependent variable does not have a transcendental function.

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It can include things like answers to equations, solutions to problems, and visualizations of data.

1. Input: The data and instructions a system receives.

2. Process: The manipulation of data and instructions to produce an output.

3. Output: The result of the system's processing of the data and instructions.

4. Feedback: Information from the output that is used to improve the system's performance.

1. Input: The first part of a system in mathematics is the input. This is the data and instructions that the system receives. It can include things like numbers, equations, formulas, and instructions for how to process the data.

2. Process: The second part of a system in mathematics is the process. This is where the data and instructions are manipulated to produce an output. This can include things like calculations, algorithms, and other mathematical operations.

3. Output: The third part of a system in mathematics is the output. This is the result of the system's processing of the data and instructions. It can include things like answers to equations, solutions to problems, and visualizations of data.

4. Feedback: The fourth part of a system in mathematics is the feedback. This is information from the output that is used to improve the system's performance. This can include things like error messages, suggested changes, and suggestions for optimization.

The four parts of a system in mathematics are input, process, output, and feedback. Input is the data and instructions a system receives. Process is the manipulation of data and instructions to produce an output. Output is the result of the system's processing of the data and instructions. Feedback is information from the output that is used to improve the system's performance.

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1. The area of the shaded part = 95π

2. The area of the half circle = 12.5π

3. the distance of the bicycle = 0.0675/π inches

4. The diameter of the circle = 9 unit

radius = 4.5

area = 20.25π unit²

A circle is simply a round shape that has no corners or line segments. It is a closed curve shape in geometry. The points of circle are at a fixed distance from the center.

The area of a circle = πr²

1. Area of the shaded part = area of big circle - area of small circle

= 12²π - 7²π

= 144π - 49π

= 95π

2. area of semi circle = 1/2πr²

= 1/2×5²× π

= 25π/2

= 12.5π unit²

3. The distance moved by the biycle =

S = tetha/r

tetha = 2π×100 = 200π

S = 200π/13.5

S = 14.8π inches

4. c = πd

c = 9π

d = 9

r =d/2 = 9/2 = 4.5

Area = 4.5²π

A = 20.25π unit²

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Answer:41,111

Step-by-step explanation:4-3=1 3-2=1 2-1=1 bring the 1 down and bring the 4down to and you got your answer

Answer:

Factors of x are 0, 5, and -4.

Step-by-step explanation:

⇒ x³ = x² + 20x

⇒ x³ - x² - 20x = 0

⇒ x (x² - x - 20) = 0

⇒ x ( x² - 5x + 4x - 20) = 0

⇒ x ( x ( x - 5) + 4 ( x - 5)) = 0

⇒ x (x - 5) (x + 4) =0

⇒ x = 0, 5, and -4

Answer: x = 5

x = -4

x = 0

Step-by-step explanation:The first term is,  x2  its coefficient is  1 .

The middle term is,  -x  its coefficient is  -1 .

The last term, "the constant", is  -20

Step-1 : Multiply the coefficient of the first term by the constant   1 • -20 = -20

Step-2 : Find two factors of  -20  whose sum equals the coefficient of the middle term, which is   -1 .

     -20    +    1    =    -19

     -10    +    2    =    -8

     -5    +    4    =    -1    That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -5  and  4

                    x2 - 5x + 4x - 20

Step-4 : Add up the first 2 terms, pulling out like factors :

                   x • (x-5)

             Add up the last 2 terms, pulling out common factors :

                   4 • (x-5)

Step-5 : Add up the four terms of step 4 :

                   (x+4)  •  (x-5)

            Which is the desired factorization  A product of several terms equals zero.

When a product of two or more terms equals zero, then at least one of the terms must be zero.

We shall now solve each term = 0 separately

In other words, we are going to solve as many equations as there are terms in the product

Any solution of term = 0 solves product = 0 as well.  Solving   x2-x-20 = 0 by Completing The Square .

Add  20  to both side of the equation :

  x2-x = 20

Now the clever bit: Take the coefficient of  x , which is  1 , divide by two, giving  1/2 , and finally square it giving  1/4

Add  1/4  to both sides of the equation :

 On the right hand side we have :

  20  +  1/4    or,  (20/1)+(1/4)

 The common denominator of the two fractions is  4   Adding  (80/4)+(1/4)  gives  81/4

 So adding to both sides we finally get :

  x2-x+(1/4) = 81/4

Adding  1/4  has completed the left hand side into a perfect square :

  x2-x+(1/4)  =

  (x-(1/2)) • (x-(1/2))  =

 (x-(1/2))2

Things which are equal to the same thing are also equal to one another. Since

  x2-x+(1/4) = 81/4 and

  x2-x+(1/4) = (x-(1/2))2

then, according to the law of transitivity,

  (x-(1/2))2 = 81/4

We'll refer to this Equation as  Eq. #4.2.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of

  (x-(1/2))2   is

  (x-(1/2))2/2 =

 (x-(1/2))1 =

  x-(1/2)

Now, applying the Square Root Principle to  Eq. #4.2.1  we get:

  x-(1/2) = √ 81/4

Add  1/2  to both sides to obtain:

  x = 1/2 + √ 81/4

Since a square root has two values, one positive and the other negative

  x2 - x - 20 = 0

  has two solutions:

 x = 1/2 + √ 81/4

  or

 x = 1/2 - √ 81/4

Note that  √ 81/4 can be written as

 √ 81  / √ 4   which is 9 / 2

Answer:

Step 1: The nth term of an arithmetic sequence is given by an = a + (n – 1)d. So, to find the nth term, substitute the given values a = 2 and d = 3 into the formula. Step 2: Now, to find the fifth term, substitute n = 5 into the equation for the nth term.

The probability that at least 3 copiers will break down on a given day is 0.0909.

1. Calculate the probability that 0 copiers will break down:

P(0) = (1-0.66)^5

= 0.0729

2. Calculate the probability that 1 copier will break down:

P(1) = 5 * 0.66 * (1-0.66)^4

= 0.2221

3. Calculate the probability that 2 copiers will break down:

P(2) = 5C2 * 0.66^2 * (1-0.66)^3

= 0.3088

4. Calculate the probability that 3 copiers will break down:

P(3) = 5C3 * 0.66^3 * (1-0.66)^2

= 0.2197

5. Calculate the probability that at least 3 copiers will break down:

P(at least 3) = 1 - (P(0) + P(1) + P(2))

= 0.0909

The probability that at least 3 copiers will break down on a given day is 0.0909. This was calculated by first calculating the probability of 0 copiers breaking down (P(0)) which is (1-0.66)^5, then the probability of 1 copier breaking down (P(1)) which is 5 * 0.66 * (1-0.66)^4, and then the probability of 2 copiers breaking down (P(2)) which is 5C2 * 0.66^2 * (1-0.66)^3. Finally, the probability that at least 3 copiers will break down (P(at least 3)) was calculated by subtracting the sum of P(0), P(1), and P(2) from 1. This gives us the final result of 0.0909.

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The rate of increase of the depth of the water is 6.67 ft/min.

First, calculate the volume of the water tank:

Volume = (1/3) πr²h

Volume = (1/3) × 3.14 × 10² × 24

Volume = 7,040 ft³

Now calculate the rate of increase of the depth of the water:

Rate of increase of the depth = (Flow rate of water into tank / Volume of tank)

Rate of increase of the depth = (20 ft³/min / 7,040 ft³)

Rate of increase of the depth = 0.0028 ft/min

Finally, calculate the rate of increase of the depth of the water when the water is 16 ft deep:

Rate of increase of the depth = (Flow rate of water into tank / Volume of water)

Rate of increase of the depth = (20 ft³/min / 4,096 ft³)

Rate of increase of the depth = 0.0049 ft/min

Therefore, the rate of increase of the depth of the water when the water is 16 ft deep is 0.0049 ft/min, which is equivalent to 6.67 ft/min.

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The measurements of all the angles, if one angle is a right angle (90 degrees), and the second angle's measurement is six times smaller than the third angle's measurement of 180 degrees.

As per the data given in the above question are as bellow,

The data provided are as bellow.

The known angle (90) a pronumeral of r.

give the second angle (the one with all the subtraction) a pronumeral or x.

The third angle the pronumeral of y

x = 7y - 6

90 + x + y = 180

90 + 7y - 6 + y = 180

90 + 8y - 6 = 180

90 + 8y =186

8y = 96

y = 12.  

the third angle is 12

so 12 + 90 = 102

the second angle must equal 88 to make it to 180

For such more question on right triangle.

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Note the correct question is as bellow,

Given a right triangle, find the measures of all of the angles, if one angle is a right angle (90 degrees) and the measure of the second angle is six less than seven times the measure of the third angle. This is represented by the equation x=7y - 6

Answer: C

Step-by-step explanation: a triangle has a measurement of 180 degrees not 120

the answer is c the measure of each angle of an Equlateral triangle in 120 °