Use the Exterior Angle Theorem to find the measure of each angle.

mangleC =
mangleD =
mangleDEC =

If the length of a rectangle is four times its width. if the area of the rectangle is 256 cm2, the perimeter of the rectangle is 80 cm.

Let's first use the given information to write two equations relating the length and width of the rectangle:

The length L is four times the width W: L = 4W

The area A is 256 cm²: A = LW = 256

Substituting equation 1 into equation 2, we get

4W^2 = 256

Solving for W, we get:

W^2 = 64

W = 8 (since we're looking for a positive value)

Substituting this value back into equation 1, we get:

L = 4W = 32

Therefore, the length and width of the rectangle are 32 cm and 8 cm, respectively.

The perimeter of the rectangle is twice the sum of its length and width:

P = 2(L + W) = 2(32 + 8) = 80 cm

So the perimeter of the rectangle is 80 cm.

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The standard deviation expressed as a percent of the mean is the D: coefficient of variation.

The coefficient of variation (CV) is a statistical measure that expresses the standard deviation of a data set as a percentage of the mean. It is particularly useful for comparing the variability of different data sets with different units or scales, and for identifying the degree of variation relative to the mean. The formula for the coefficient of variation is:

CV = (standard deviation / mean) x 100%

So, the coefficient of variation is a dimensionless quantity that measures the relative dispersion of the data.

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the graph of the function is attached below.

A relationship between a group of inputs and one output is referred to as a function. In plain English, a function is an association between inputs in which each input is connected to precisely one output. A domain, codomain, or range exists for every function. Typically, f(x), where x is the input, is used to represent a function.

Given a function,  f(x) = 3√x -2

To graph the function f(x) = 3√x -2, we can follow these steps:

f(0) = 3√0 - 2 = -2

f(1) = 3√1 - 2 = 1

f(8) = 3√8 - 2 ≈ 4.9

f(27) = 3√27 - 2 ≈ 7.2

Here, the graph of f(x) = 3√x -2 is attached below.

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Answer:

$2.79

Step-by-step explanation:

We know

Peter bought a sandwich for $4.25, a drink for $2.17, and a cookie for $0.79.

4.25 + 2.17 + 0.79 = $7.21

If he paid with a $10 bill, how much change did Peter get?

10 - 7.21 = $2.79

So, Peter get $2.79 in change.

The speed of the fire is 3.214 miles per hour (rounded to three decimal places)

It is defined as the distance traveled by an object per unit of time, typically in meters per second (m/s) or kilometers per hour (km/h).

Speed can be calculated by dividing the distance traveled by the time taken to cover that distance. It is a scalar quantity, meaning it only has magnitude and no direction.

Let's call the horizontal distance the fire advanced during the 6 minutes "d." We can use the tangent function to find d:

tan(12.3°) = (3334 - h) / d

tan(10.5°) = (3334 - h) / d

Solving for d in each equation, we get:

d = (3334 - h) / tan(12.3°)

d = (3334 - h) / tan(10.5°)

Since the time interval is the same for both observations, the fire advanced a distance equal to the difference between the two distances:

d = [(3334 - h) / tan(12.3°)] - [(3334 - h) / tan(10.5°)]

Plugging in the values and simplifying:

d = 1698.5 feet (rounded to the nearest tenth)

To find the speed at which the fire is spreading towards the ranger, we need to convert the distance from feet to miles, and the time from minutes to hours:

d = 1698.5 feet / 5280 feet per mile = 0.3214 miles

t = 6 minutes / 60 minutes per hour = 0.1 hours

The speed of the fire is then:

s = d / t = 0.3214 miles / 0.1 hours = 3.214 miles per hour (rounded to three decimal places)

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The required interval of domain for the composite function is  [-2, 3].

The range of values that we are permitted to enter into our function is known as the domain of a function. The x values for a function like f make up this set (x). A function's range is the collection of values it can take as input. After we enter an x value, the function outputs this sequence of values.

The domain of fog(x) is the set of all values of x for which the composition function fog(x) is defined.

fog(x) means that we plug g(x) into f(x), so we have:

fog(x) = f(g(x)) = f(x^2 - x) = √(6 - (x^2 - x))

= √(6 - x^2 + x)6

For the expression under the square root to be defined, we must have 6 - x^2 + x ≥ 0. This is a quadratic inequality that can be factored as:

-(x - 3)(x + 2) ≤ 0

The solutions of this inequality are -2 ≤ x ≤ 3.

However, we also need to check that the expression under the square root is non-negative, so we need to exclude the values of x that make 6 - x^2 + x < 0. This inequality holds for x < -2 or x > 3.

Therefore, the domain of fog(x) is [-2, 3].

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The correlation between X and Y is -0.648, rounded to two decimal places.

First, we need to calculate the means of X and Y:

mean(X) = (1 + 1.5 + 1.5 + 2.5 + 3) / 5 = 2

mean(Y) = (1/4 + 1/8 + 1/4 + 1/4 + 1/8) / 5 = 0.15

Next, we can calculate the covariance using the formula:

cov(X, Y) = E[(X - mean(X)) * (Y - mean(Y))] = Σ[(X - mean(X)) * (Y - mean(Y))] / (n - 1)

where n is the number of data points. Substituting the values, we get:

cov(X, Y) = [(1 - 2) * (1/4 - 0.15) + (1.5 - 2) * (1/8 - 0.15) + (1.5 - 2) * (1/4 - 0.15) + (2.5 - 2) * (1/4 - 0.15) + (3 - 2) * (1/8 - 0.15)] / 4

= -0.035

Therefore, the covariance between X and Y is -0.035.

Finally, we can calculate the correlation coefficient using the formula:

corr(X, Y) = cov(X, Y) / (stddev(X) * stddev(Y))

where stddev is the standard deviation. We can calculate the standard deviations as follows:

stddev(X) = sqrt([Σ(X - mean(X))^2] / (n - 1)) = 0.8292

stddev(Y) = sqrt([Σ(Y - mean(Y))^2] / (n - 1)) = 0.0632

Substituting the values, we get:

Corr(X, Y) = -0.035 / (0.8292 * 0.0632) = -0.648

Therefore, the correlation between X and Y is -0.648, rounded to two decimal places.

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The equation of the plane passing through P1(12,3), P2(3,2,1), and perpendicular to the plane 4−y+2z=7 is -3x + 51y - 8z = -39.

To find a plane that passes through the points P1(12,3) and P2(3,2,1) and is perpendicular to the plane 4−y+2z=7, we can follow these steps:

Find the normal vector of the given plane by taking the coefficients of x, y, and z in the equation 4−y+2z=7. The normal vector is (1, -1, 2), because the coefficients of x, y, and z are 1, -1, and 2 respectively.

Find the direction vector of the line passing through P1 and P2 by subtracting the coordinates of P1 from P2. The direction vector is (-9, -1, -2), because P2 - P1 = (3-12, 2-3, 1-0) = (-9, -1, -2).

Find the cross product of the normal vector of the given plane and the direction vector of the line passing through P1 and P2 to get the normal vector of the plane we are looking for. The cross product is:

(1, -1, 2) x (-9, -1, -2) = (-3, -17, -8)

Plug one of the given points, say P1(12,3), and the normal vector we found in step 3 into the point-normal form of the equation of a plane:

-3(x - 12) - 17(y - 3) - 8(z - 0) = 0

Simplifying, we get:

-3x + 51y - 8z = -39

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The two-column method table to prove that the triangles ΔLMN and ΔLJK are similar, ΔLMN ~ ΔLJK, where LM/LJ = LN/LK can be completed as follows

Statements             [tex]{}[/tex]                         Reasons

LM/LJ = LN/LK     [tex]{}[/tex]                            Given

∠L ≅ ∠L [tex]{}[/tex]                                          Reflexive property of congruency

ΔLMN ~ ΔLJK [tex]{}[/tex]                                 SAS similarity theorem

Similar triangles are triangles that have the same shape, and interior angles, but which may have different side lengths such that the proportion of the corresponding sides are equivalent.

The details of the reasons used to prove the similarity of triangles ΔLMN and ΔLJK can be presented as follows;

Reflexive property of congruency

The reflexive property of congruency states that a figure, such as an angle or a length is congruent to itself

SAS similarity theorem

The SAS, Side-Angle-Side similarity theorem states that if two sides in one triangle are proportional to two sides in another triangle, and the included angle between the two sides in both triangles are congruent, then the two triangles are similar.

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Answer:3x^3−4x^2+2x+6

Step-by-step explanation:

1).subtract the numbers

2).combine like terms

Answer: 8/9

Step-by-step explanation: 3 x 16 = 48 ounces

48 / 54 = 8/9

Answer:

Infinitely many solutions

Step-by-step explanation:

Given the equation:

[tex]\displaystyle{-16+n+14n=-16+15n}[/tex]

Which can be simplified (for left side) to:

[tex]\displaystyle{-16+15n=-16+15n}[/tex]

Both sides are same, meaning that there are infinitely many solutions since no matter which n-values you substitute in, you'll end up getting true equation.

The probability that X is between 90 and 110 = 0.2 if a normally distributed random variable x has a mean of 100 and P(x < 90) = 0.40.

The majority of the observations are centered around the middle peak of the normal distribution, which is a continuous probability distribution that is symmetrical around its mean.

The probabilities for values that are farther from the mean taper off equally in both directions. Extreme values in the distribution's two tails are likewise rare. Not all symmetrical distributions are normal, even though the normal distribution is symmetrical.

A normally distributed random variable x has a mean of 100

P(x < 90) = 0.4

The probability that X is between 90 and 110

Mean is 100

Hence P (< 100) = 0.5  as Mean is centered

Hence

P  (90 < X < 100) = 0.5 - 0.4 = 0.1

The difference between 100 and 90 is 10

The difference between 100 and 110 is 10

and 100 is Mean

Hence P  (100 < X < 110) = P  (90 < X < 100) = 0.1

Adding Both

P ( 90 < X < 110) = 0.1 + 0.1 = 0.2

The probability that X is between 90 and 110 = 0.2

Using z score table.

Z  = ( Value - Mean) /SD

Z score for 0.4 = -0.254

-0.253 = (90 - 100)/SD

SD ≈ 39.53

Z = ( 110 - 100)/39.53

Z = 0.253

For z score  < 0.253, the probability is 0.6

Hence Probability in Between is 0.6 - 0.4 = 0.2

The probability that X is between 90 and 110 = 0.2

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The number of cells in the Petri dish after 10 hours is 2¹⁰ = 1024 cells in the dish.

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

Here,
The number of cells in the Petri dish after each hour can be represented by the expression 2ⁿ, where n is the number of hours that have passed.

So, after 1 hour, there are 2¹ = 2 cells in the dish.

After 2 hours, there are 2² = 4 cells in the dish.

After 3 hours, there are 2³ = 8 cells in the dish.

And so on.

Therefore, after 10 hours, there are 2¹⁰ = 1024 cells in the dish.

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Answer:

10

Step-by-step explanation:

20% = 0.2

What is 20% of 50?

We take

50 x 0.2 = 10

So, 20% of 50 is 10

The numbers of cups that macaroni will he use will he use if he wishes makes 27 bowls is 9 cups.

If Mike makes 27 bowls of macaroni salad, we can find out how many cups of macaroni he will need by multiplying the amount of macaroni needed for one bowl by the number of bowls:

= 1/3 cup of macaroni per bowl x 27 bowls

= 9 cups of macaroni

Therefore, Mike will need 9 cups of macaroni to make 27 bowls of macaroni salad.

Full question "Mike is making macaroni salad. For each bowl of macaroni salad, he needs 1/3 cup of macaroni. How many cups of macaroni will he use will he use if he makes 27 bowls of?"

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The probability of receiving no e-mails during an hour is approximately 0.0067, or 0.67%. The probability of receiving at least three e-mails during an hour is approximately 0.8754, or 87.54%. The expected number of e-mails received during 15 minutes is 1.25. The probability of receiving no e-mails during 15 minutes is approximately 0.7788, or 77.88%

We know that the average number of e-mails received per hour is 5. Therefore, the parameter λ of the Poisson distribution is also 5, since the Poisson distribution's mean and variance are both equal to λ.

The probability of receiving no e-mails during an hour (or during any other fixed time interval of length t) can be calculated using the Poisson distribution as follows:

P(X = 0) = e^(-λ) * λ^0 / 0! = e^(-5) * 5^0 / 0! ≈ 0.0067

Therefore, the probability of receiving no e-mails during an hour is approximately 0.0067, or 0.67%.

The probability of receiving at least three e-mails during an hour can be calculated as follows:

P(X ≥ 3) = 1 - P(X < 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2))

We already know the value of P(X = 0) from part (a), so we just need to calculate P(X = 1) and P(X = 2) using the Poisson distribution:

P(X = 1) = e^(-5) * 5^1 / 1! ≈ 0.0337

P(X = 2) = e^(-5) * 5^2 / 2! ≈ 0.0842

Substituting these values into the equation above, we get:

P(X ≥ 3) = 1 - (0.0067 + 0.0337 + 0.0842) ≈ 0.8754

Therefore, the probability of receiving at least three e-mails during an hour is approximately 0.8754, or 87.54%.

Since the expected number of e-mails received per hour is 5, the expected number of e-mails received during 15 minutes (i.e., a quarter of an hour) is:

E(X) = λ * t = 5 * (1/4) = 1.25

Therefore, the expected number of e-mails received during 15 minutes is 1.25.

The probability of receiving no e-mails during 15 minutes can be calculated using the Poisson distribution with λ = 1.25 and t = 1/4:

P(X = 0) = e^(-λt) * (λt)^0 / 0! = e^(-1.25/4) * (1.25/4)^0 / 0! ≈ 0.7788

Therefore, the probability of receiving no e-mails during 15 minutes is approximately 0.7788, or 77.88%.

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The measure of variation include range, variance, iqr etc

In statistics, a measure of variation is a numerical value that describes how spread out or dispersed a set of data is. The most common measures of variation are:

Range: The range is the difference between the maximum and minimum values in a data set. It gives an idea of the spread of the data, but is sensitive to outliers.

Interquartile range (IQR): The IQR is the range of the middle 50% of the data. It is less sensitive to outliers than the range.

Variance: The variance is the average of the squared differences of each data point from the mean. It measures how much the data is spread out from the mean.

Standard deviation: The standard deviation is the square root of the variance. It is a common measure of variation and indicates the typical amount that each data point deviates from the mean.

Coefficient of variation: The coefficient of variation is the ratio of the standard deviation to the mean, expressed as a percentage. It is used to compare the variation of data sets with different means.

An overview was given due to incomplete information.

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(a) - A runs the race at a constant speed, never speeding up or slowing down. B accelerates throughout the race, starting out slower than A and, by the end, running faster than A.

(b) - Based on the graph, it appears that they are furthest apart after 8 seconds, when they are approximately 30 meters apart.

(c)- The two graphs appear to have the same slope (i.e., velocity) 9 or 10 seconds into the race.

graph attached below,

constant speed

When an object travels the same distance in the same period of time, it is said to be traveling at a constant speed. At constant speed, an object travels a uniform distance in an equal interval of time. The equation of the speed can be given as: S = d t.

Slope

Slope is a measure of the steepness of a line.

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The vectorial equation of the line that passes through (4, 0, - 3) and is parallel to the vector <2, - 4, 6> is equal to (x, y, z) = (4, 0, - 3) + t · (2, - 4, 6).

According to linear algebra, three-dimension lines can be derived from the knowledge of a point and a vector slope, whose formula is introduced below:

(x, y, z) = P(x, y, z) + t · M(x, y, z)

Where:

Two lines are parallel when they have the same vector slope. If we know that P(x, y, z) = (4, 0, - 3) and M(x, y, z) = (2, - 4, 6), then the vectorial equation of the line is:

(x, y, z) = (4, 0, - 3) + t · (2, - 4, 6)

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The following sets of functions are subspaces of v, if v be the (real) vector space of all functions f from r into r is: all f such that f(x²) = f(x²), all f which are continuous.

V is a Vector- Space of all real- valued functions over field of real numbers R and W consists of all real- valued even functions which are bounded also as a subset of V.

Let f , g belong to W then f , g both are even & bounded. Hence ;

(1) (f + g ) is even & bounded because ,

(f + g )(-x ) = f(- x )+ g(-x) = f( x) + g( x )

=( f+g)( x) and for all x€ R and

= | f (x) + g(x) | </= |f (x) | + | g (x) |

</= (c +d) = C(constant)

and similarly for any scalar k€ R , (kf ) will be an even function and it will be bounded also.

A set whose elements, frequently termed vectors, can be added to and multiplied ("scaled") by figures known as scalars is referred to as a vector space (also known as a linear space). Real numbers make up scalars most of the time, but they can also be complex numbers or, more broadly, components of any field. Certain conditions, referred to as vector axioms, must be met by the operations of vector addition and scalar multiplication.

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The pairs of linear equations that are parallel are:

2 and 4.

Two linear equations:

y = a*x + b

y = c*x + d

Are parallel if the two slopes are equal and the y-intercepts are different, then:

a = c

b ≠ d

Let's write all the lines in the slope-intercept form:

1) 4x + 3y = 15

  3y = 15 - 4x

   y = (-4/3)*x + 15/3

   y = (-4/3)*x + 5

2) 3x - 4y = -8

  -4y = -8 - 3x

     y = 2 + (3/4)x

So we can see that lines 2 and 4 are parallel.

If we rewrite line 3 we will get:

y + 1 = (4/3)*(x - 6)

y = (4/3)*x - 8  -1

y = (4/3)*x - 9

Then we conclude that only lines 2 and 4 are parallel.

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The notation for this sample is denoted by [tex]\overline{x}[/tex].

The entire group about whom you want to make conclusions is referred to as a population.

The particular group from which you will gather data is known as a sample. The sample size is always smaller than the population as a whole.

Given, In a survey of 1917 cell phone users, the mean was 13.10 phone calls a day.

This a sample so the notation for the mean is [tex]\overline{x}[/tex].

If it is for population then the notation of mean would have been [tex]\mu[/tex].

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Based on the information, we can infer that the range is an adequate indicator to affirm that the price difference of the Big Mac in different countries is minimal.

To find the range of the given data we must perform the following procedure:

We must find the highest and lowest data, in this case the highest and lowest data are 6.8 and 1.98 respectively. So, to find the range we must find the difference between these two data:

6.8 - 1.89 = 4.91

To find the variance we must apply the following formula:

S² = 1 / 11 - 1 ( 205.59 - (44.67)²/11)

S² = 2.42

To find the standard deviation we must apply the following formula:

S = [tex]\sqrt[n]{variance}[/tex]

S = [tex]\sqrt{2.42}[/tex]

S = 1.56

Based on the above data, we can infer that the range is an indicator that the difference in the Big Mac price in different countries is minimal.

Note: This question is incomplete. Here is the complete information:

Attached image

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The probability that the last person to board sits in the assigned seat is always 1/2, regardless of the number of people n in the line.

The issue can be tackled utilizing numerical enlistment.

Base Case:

At the point when n = 2, there are two travelers and two seats. The principal traveler takes an irregular seat, and the subsequent traveler will sit in his doled out seat with likelihood 1/2 or take the other seat with likelihood 1/2. Consequently, p2 = 1/2.

Inductive Speculation:

Accept that for some certain number k, pk = 1/2.

Inductive Step:

Think about n = k+1 travelers. The principal traveler takes an irregular seat. There are two cases to consider:

Case 1: The primary traveler sits down relegated to the k+1-th traveler. For this situation, the last traveler will be ensured to sit in his doled out seat, and the leftover k travelers can be considered as a subproblem with similar circumstances. By the inductive speculation, the likelihood that the last traveler in the subproblem sits in his alloted seat is 1/2. Thusly, the likelihood that the last traveler in the first issue sits in his doled out seat for this situation is 1/(k+1) + (k/(k+1)) * 1/2 = (k+2)/(2(k+1)).

Case 2: The primary traveler sits down other than the one doled out to the k+1-th traveler. For this situation, the issue lessens to the case with k travelers, and the likelihood that the last traveler in the decreased issue sits in his doled out seat is pk = 1/2 (by the inductive speculation). Thusly, the likelihood that the last traveler in the first issue sits in his doled out seat for this situation is (k/(k+1)) * 1/2 = k/(2(k+1)).

Joining the two cases, we have:

pk+1 = (k+2)/(2(k+1)) + k/(2(k+1)) = (k+1)/(2(k+1)) = 1/2.

Consequently, by numerical acceptance, the likelihood that the last individual to board sits in the appointed seat is dependably 1/2, no matter what the quantity of individuals n in the line.

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Answer:

1. 40

2. 351

3. 49

Step-by-step explanation:

just multiply the 2 numbers!

5 times 8 is 40

13 times 27 is 351

7 times 7 is 49

(square is the same on every side so every side is 7)

The ordered pair that doesn't represent a function is: D.. {(-1,7), (7, 1), (5,-7), (7,-8)}.

A function is a mathematical concept that associates an input (or argument) to a unique output (or value). In mathematical notation, a function is represented as f(x), where x is the input, and f(x) is the output. The association between the input and output is defined by a set of rules, or a formula, that assigns a unique output value to each input value

In a function, for each unique x-value, there must be exactly one corresponding y-value.

D. {(-1,7), (7, 1), (5,-7), (7,-8)} does not represent a function because the same x-value (7) corresponds to two different y-values (1 and -8).

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Please find attached the graph of the function of the length with respect to the width of the rectangle, created with MS Excel, where, 1 ≤ W ≤ 11, and the perimeter of the rectangle is 24 cm.

A graph of a function, is a representation of the ordered pairs of the points of the function, (x, f(x)), on the coordinate plane.

Whereby the figure is a rectangle, we get;

Perimeter = 2 × Length of the rectangle + 2 × Width of the rectangle

Let P represent the perimeter of the rectangle and let L represent the length of the rectangle and let W represent the width of the rectangle, we get;

P = 2·L + 2·W

The perimeter of the rectangle. P = 24

Therefore;

24 = 2·L + 2·W

2·L = 24 - 2·W

L = (24 - 2·W)/2 = 12 - W

L = 12 - W

The graph of the above linear equation representing the length of the rectangle, L, where the width, W is; 1 ≤ W ≤ 11.

Please find attached the required graph of length, L to the width W of the rectangle created with MS Excel

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The four points on the graph of this function f(x) = √(x + 1) + 4 has been plotted on the graph shown in the image attached below.

In Mathematics, a graph can be defined as a type of chart that is typically used for the graphical representation of data points or ordered pairs on both the horizontal and vertical lines of a cartesian coordinate, which are the x-axis and y-axis respectively.

Next, we would use an online graphing calculator to plot the given absolute value function f(x) = √(x + 1) + 4 as shown in the graph attached below.

By observing critically the graph (see attachment), the four ordered pairs (points) that fit on its axes include the following:

Ordered pair = (-1, 4), which is the leftmost.

Ordered pair = (0, 5).

Ordered pair = (3, 6).

Ordered pair = (8, 7).

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The height of the triangle is 3/5.

A triangle is said to be right-angled if one of its inner angles is 90 degrees, or if any one of its angles is a right angle.

Given:

A right-angled triangle.

Let the values of the hypotenuse in parts be m and n.

So,

(1)² = (5/4)m

m = 4/5

And n = 9/16 x 4/5

n = 9/20

So, the height of the triangle is,

h² = 9/20 x 4/5

h² = 9/25

h = 3/5

Therefore, h = 3/5.

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