kieran tried to shade all the multiples of 3 on the hundred chart, but he made a mistake. what mistake did kieran make?

Answer:

52°

Step-by-step explanation:

This is an equilateral triangle. Therefore all the sides and angles are equal to each other.

We know that,

the sum of the interior angles in a triangle is 180° and in equilateral triangles each angle measured 60° ( 180°/3 ).

Accordingly,

x + 8 = 60

Subtract 8 from both sides.

x = 60 - 8

x = 52°

4

Step-by-step explanation:

[tex]{ \tt{ \frac{5}{1} \times \frac{4}{5}}} [/tex]

[tex]{ \tt{ \frac{5 \times 4}{5}}} [/tex]

[tex]{ \tt{ = \frac{20}{5}}} [/tex]

[tex] = { \green{ \boxed{ \pink{ \sf{4}}}}}[/tex]

Answer:

no

Step-by-step explanation:

the percentage has nothing to do with how many of them there are, forgive me if im wrong

The exact number of books on the shelf cannot be determined based on the information given.

An expression is a grouping of one or more mathematical or logical operators, operands (values, variables, or other expressions), and brackets in mathematics and computer programming that denote a computation that can be evaluated to generate a value.

We know that between 50 and 100 books are stored on a shelf, so let's assume the number of books is some integer value between 50 and 100, inclusive. Let's call this value "n".

Exactly 20% of the books are textbooks, which means there are 0.2n textbooks on the shelf.

Exactly one-seventh of the books are novels, which means there are (1/7)n novels on the shelf.

We know that n, 0.2n, and (1/7)n are all integers because they represent whole numbers of books. We can write this as:

n = a

0.2n = b

(1/7)n = c

where a, b, and c are integers.

From the second equation, we know that 0.2n is a multiple of 1/10. From the third equation, we know that n is a multiple of 7. Therefore, we can rewrite these equations as:

n = 10b

n = 7c

Since n is equal to both 10b and 7c, we know that n must be a multiple of the least common multiple of 10 and 7, which is 70.

So, n must be a multiple of 70, and therefore must be one of the integers 50, 70, or 90. However, we cannot determine which of these three values is the correct one based on the information given, because all three satisfy the conditions in the problem:

If n=50, then 20% of the books (i.e., 10 books) are textbooks, and 1/7 of the books (i.e., 7.14 books, which we round down to 7) are novels.

If n=70, then 20% of the books (i.e., 14 books) are textbooks, and 1/7 of the books (i.e., 10 books) are novels.

If n=90, then 20% of the books (i.e., 18 books) are textbooks, and 1/7 of the books (i.e., 12.86 books, which we round down to 12) are novels.

Therefore, the exact number of books on the shelf cannot be determined based on the information given.

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The actual end-of-year payment or present value of this ordinary annuity is equal to $1,482.

For the rate of interest, we have:

Rate of interest, r = APR/12

Rate of interest, r = 0.12/12

Rate of interest, r = 0.01

Next, we would calculate the present value of this ordinary annuity by using this formula:

Present value, PV = [P ÷ [1 - (1 + r)^{-n}]/r]

Where:

Substituting the given parameters into the formula, we have;

Present value, PV = 4,500 ÷ ((1 - (1 + 0.12)^(-4))/(0.12))

Present value, PV = 4,500 ÷ ((1 - (1.12)^(-4))/(0.12))

Present value, PV = 4,500 ÷ ((1 - 0.635518078)/(0.12))

Present value, PV = 4,500 ÷ ((0.364481922)/(0.12))

Present value, PV = 4,500/3.0373

Present value, PV = $1,481.58 ≈ $1,482.

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The values of the variables, x and y in the angles formed by the parallel lines that have a common transversal are;

x = 4°, y = 10°

11. x = 9°, y = 12°

12. x = 14°, y = 5°

13. x = 8°, y = 11°

14. x = 16°, y = 23°

15. x = 9°, y = 13°

Parallel lines are lines that maintain the same distance from each other along their lengths.

From the figure, we get;

(29·x - 3) and (15·x + 7) are linear pair angles, therefore;

(29·x - 3) + (15·x + 7) = 180°

44·x + 4 = 180

x = (180 - 4)/44 = 4

x = 4°

(15·x + 7) and (13·y - 17) are same side exterior angles between parallel lines, therefore;

(15·x + 7) + (13·y - 17) = 180°

(15×4 + 7) + (13·y - 17) = 180°

67 + 13·y - 17 = 180°

50 + 13·y = 180

13·y = 180 - 50 = 130

y = 130 ÷ 13 = 10

y = 10°

11. (18·x - 44)° and  (8·x - 10)° are same side exterior angles, therefore;

(18·x - 44)° +  (8·x - 10)° = 180°

26·x - 54 = 180

x = (180 + 54)/26 = 9

x = 9°

(8·x - 10)° and (13·y - 38)° are linear pair angles, therefore;

(8·x - 10)° + (13·y - 38)° = 180°

(8 × 9 - 10)° + (13·y - 38)° = 180°

62 - 38 + 13·y = 180

13·y = 180 - (62 - 38) = 156

y = 156 ÷ 13 = 12

y = 12°

12. (9·x - 2)° and (5·x + 54)° are corresponding angles, therefore;

(9·x - 2)° = (5·x + 54)°

(9·x - 2)° = (5·x + 54)°

9·x - 5·x = 54 + 2 = 56°

4·x = 56°

x = 56° ÷ 4 = 14°

x = 14°

(9·x - 2)° and (10·y + 6)° are linear pair angles, therefore;

(9·x - 2)° + (10·y + 6)° = 180°

(9 × 14 - 2)° + (10·y + 6)° = 180°

124 + 6 + 10·y = 180

10·y = 180 - 130 = 50

y = 50 ÷ 10 = 5

y = 5°

13. The angle to which the angle (8·x - 1)° is an alternate interior angle is a corresponding angle to the angle (11·x - 25)°, therefore;

(8·x - 1)° is congruent to angle (11·x - 25)°

(8·x - 1)° = (11·x - 25)°

11·x - 8·x = 25 - 1 = 24

3·x = 24

x = 24 ÷ 4 = 8

x = 8°

Angles (15·y - 48)° and (8·x - 1)° are linear pair angles, therefore;

(15·y - 48)° + (8·x - 1)° = 180°

(15·y - 48)° + (8 × 8 - 1)° = 180°

(15·y - 48)° + 63° = 180°

15·y + 15° = 180°

15·y = 180° - 15° = 165°

y = 165° ÷ 15 = 11°

y = 11°

14· The angle (4·x + 4)° and (7·x - 44)° are alternate exterior angles, therefore;

(4·x + 4)° = (7·x - 44)°

7·x - 4·x = 44° + 4° = 48°

3·x = 48°

x = 48° ÷ 3 = 16°

x = 16°

The vertical angle to angle 39° is a linear pair angle to the angle (8·y - 43)°

Therefore;

39 + 8·y - 43 = 180°

8·y  = 180 + 43 - 39 = 184

y = 184 ÷ 8 = 23

y = 23°

15. The diagram indicates that the angles (15·x - 26)° and (12·x + 1)° are congruent, therefore;

(15·x - 26) = (12·x + 1)

(15·x - 12·x) = (26 + 1)

3·x = 27

x = 27 ÷ 3 = 9

x = 9°

The vertical angle to (2·x + 1) is therefore;

12·x + 1 = 12 × 9 + 1 = 109

The angle sum property indicates that we have;

28° + 109° + (4·y - 9)° = 180°

(4·y - 9)° = 180° - (28° + 109°) = 43°

(4·y - 9)° = 43°

(4·y)° = 43° + 9° = 52°

y = 52°/4 = 13°

y = 13°

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Answer:

  3a.  $8493.42

  3b.  139.0 months

  4a.  10

  4b.  10.2 years

Step-by-step explanation:

Given exponential equations, you want values for specific times, and you want times for specific values.

In part (a), the value is found by substituting an appropriate value for t and doing the arithmetic.

In part (b), logarithms will be involved in solving for t.

(a) Use t=12, the number of months in one year. You want the value of ...

  S = 8000·1.005^12 ≈ 8493.42

The amount after 1 year is $8493.42.

(b) The time it takes to double the investment is the value of t such that ...

  16000 = 8000·1.005^t

  2 = 1.005^t . . . . . . divide by 8000

  log(2) = t·log(1.005) . . . . . take logarithms

  t = log(2)/log(1.005) ≈ 138.976

The investment will double in about 139.0 months.

(a) Use t=0 and evaluate.

  y = 2500 -2490e^(-0.1·0) = 2500 -2490 = 10

The population of otters was 10 when they were reintroduced.

(b) The time it takes for the population to reach 1600 is the value of t such that ...

  1600 = 2500 -2490e^(-0.1t) . . . . use 1600 for y

  -900 = -2490e^(-0.1t) . . . . . . . . . subtract 2500

  900/2490 = e^(-0.1t) . . . . . . . . . divide by -2490

  ln(900/2490) = -0.1t . . . . . . . . . take natural logs

  t = ln(900/2490)/-0.1 . . . . . . . . divide by the coefficient of t

  t ≈ 10.176 ≈ 10.2

It will be 10.2 years before the otter population reaches 1600.

The exact arc length of f(x)=√2x+1 is 4.5.

The arc length is the separation between two points along a curve segment. The arc length of a function f(x) is given by the formula,

[tex]L=\int\limits^a_b {\sqrt{1+f'(x)^2}} \, dx[/tex]

Where f'(x) is derivative of function f(x). The arc length is, to put it simply, the distance that passes across the curved line of the circle that forms the arc. It should be noted that the arc's length is greater than the separation of its ends along a straight line.

Finding the arc length of f(x)=√2x+1 0≤x≤4, using the formula,

First finding the derivative of function,

[tex]f'(x)=\frac{d(\sqrt{2x+1})}{dx} \\\\=\frac{2}{2\sqrt{2x+1}} \\\\=\frac{1}{\sqrt{2x+1}}[/tex]

Now, putting the derivative of function f(x) in the formula of arc length L,

[tex]L=\int\limits^a_b {\sqrt{1+f'(x)^2}} \, dx\\\\L=\int\limits^4_0 {\sqrt{1+(\frac{1}{\sqrt{2x+1}})^2}} \, dx\\\\L=\int\limits^4_0 {\sqrt{1+\frac{1}{2x+1}}}\\\\L=\int\limits^4_0 {\sqrt{\frac{2x+1+1}{2x+1}}}\\L=\int\limits^4_0 {\sqrt{\frac{1}{2x+1}}}\\\approx4.5[/tex]

Therefore, the exact arc length of f(x)=√2x+1 is 4.5.

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The winner of the race among the friends would be Leslie.

The person who has the highest average speed will win the race. Average speed is the total distance travelled per time.

Average speed = total distance / total time

Carolina's average speed = 7/8 ÷ 1/2

7/8 x 2 = 7/4 = 1 3/4 miles per hour

Leslie's average speed = 1 ÷ 1/3

1 x 3 = 3 miles per hour

Bryan's average speed = 3/4 ÷ 3/4

3/4 x 4/3 = 1 mile per hour

Javier's average speed = 1 ÷ 2/3

1 x 3/2 = 1 1/2 miles per hour

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The width across Keala book cover is 40cm

A rectangle is a type of quadrilateral, whose opposite sides are equal and parallel. It is a four-sided polygon that has four angles, equal to 90 degrees

From the question,

The length of the cover

= 22 1/4

= 89/4

The area of the cover

=890 cm²

From the formula

Area of a Rectangle = Length x Breadth

890=89/4×Breadth

Breadth = 890×4/89

Breadth=40cm

Hence, the width of the book cover is 40cm

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The factors of 6x²y-10xy+15x-25​ are (3x-5)(2xy+5).

According to the question,

We have the following expression:

6x²y-10xy+15x-25​

In this expression, we can not use any identity to solve it further.

Now, in order to find the factors of this expression, we will find the common terms from the first two terms and one common factor from the last two terms.

Taking 2xy as a common factor from first two terms and 5x as a common factor from the last two terms:

2xy(3x-5)+5(3x-5)

Taking (3x-5) as a common factor from the above expression:

(3x-5) (2xy+5)

Hence, the factors of 6x²y-10xy+15x-25​ are (3x-5)(2xy+5).

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The missing side length of  triangle XYZ is 7.

Given,

Triangle ABC and triangle XYZ are similar.

Length of AB = 4

Length of BC = 8

Length of AC = x

Length of XY = 24

Length of YZ = 48

Length of XZ = 42

We have to find the missing length x.

Here,

Triangle ABC and triangle XYZ are similar.

So,

AB : XY = BC : YZ = AC : XZ

Then,

Take AB : XY = AC : XZ

= 4 : 24 = x : 42

= 4 × 42 = 24x

x = 4 × 42 / 24

x = 42/6

x = 7

That is,

The missing side length of  triangle XYZ is 7.

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Answer: 119

Step-by-step explanation:

(4^2+1)(3*4-5)

=(16+1)(12-5)

=17*7

=119

Answer:

g(4) = 119

Step-by-step explanation:

g(x) = (x² + 1) (3x - 5)

Substitute x for 4 in the equation

g(4) = (4² + 1) (3(4) - 5)

g(4) = (16 + 1) (12 - 5)

g(4) = (17) (7)

g(4) = 119

Hence, the value of g(4) is 119

The fifth term of the geometric progression is -0.125 and the sum of the first 20 terms is  [tex]\frac{4}{3}(\frac{1-2^{20} }{2^{20} }) }[/tex]

A geometric progression or a geometric sequence is the sequence, in which each term is varied by another by a common ratio.

All terms of a geometric progression have a common multiple called the common ratio.

if a = -2, which is first term

r = -0.5 which is common ration,

Then fifth term = a[tex]r^{4}[/tex]

substitute a and r into the above equation, you have

fifth term = (-2)([tex](-0.5)^{4}[/tex] = - 0.125

sum of G.P when the common ratio is less than 1, we have

[tex]\frac{a(1 - r^{n} )}{1 - r}[/tex]

where n = number of terms = 20

sum = [tex]\frac{-2(1 - (-0.5)^{20}) }{1--0.5}[/tex] =  [tex]\frac{4}{3}(\frac{1-2^{20} }{2^{20} }) }[/tex]

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The probability that the proportion of defective bottles in a sample of 546 bottles would differ from the population proportion by greater than 4% is: 10.9069.

To obtain the correct value, we need to use the formula for the Z score. This formula requires that we subtract the average or mean value from the data point provided and divide the remaining value by the standard deviation.

Here is the formula:

z = (x-μ)/σ

First population proportion = 0.24

The sample proportion, hat p = 0.04

​Z score =

[tex]0.04 - 0.24 / \sqrt 0.24 (1 - 0.24) / 546[/tex]

So, the Z score will

= -10.9469

So, when the z score = - 10.9469, the probability that Z is greater than - 10.9469,  is = 10.9069

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Based on the given residual plot, as regards the linear model, we can say that; A linear model is not suitable because the residual plot shows a curved pattern.

A residual plot is a plot that shows the difference between the observed response and the fitted response values.

The ideal residual plot is usually called the null residual plot, and it shows a random scatter of points that form an approximately constant width band around the identity line.

When it comes to finding out if a linear model will be suitable, we can say that the points will form a pattern when the model function is not correct.

Now, you might be able to transform variables or even add polynomial and interaction terms in order to remove the pattern.

Now, looking at the given graph, we can say that a linear model is not suitable because from the definitions above, we can see that the residual plot shows a curved pattern.

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The solution of the given statement will be inequality c > -60.

Inequality is defined as mathematical statements that have a minimum of two terms containing variables or numbers is not equal.

We have been given the statement "number c divided by 3 is greater than -20".

As per the given statement, we can represent this algebraic form with inequality as:

⇒ c / 3 > -20

Inequality by multiplying both sides of the inequality by 3:

⇒ 3 × c / 3 > 3 × -20

⇒ c > -60

Thus, the solution of the given statement will be inequality c > -60.

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The question seems to be incomplete the correct question would be

Find the solution if a number c divided by 3 is greater than -20.

Answer:

103 N

Step-by-step explanation:

we use the equation: w = mg

where m is mass, W is weight and g is gravitational field strength

w = ?

m= 10.5 kg

g = 9.81 ms‐² (for earth)

so,

w = 10.5 × 9.81

w = 103.005

w ≈ 103 Newtons

Answer:

see explanation

Step-by-step explanation:

(a)

perimeter (P) is calculated as

P = 2(l + b) ← l is the length and b the breadth

given l = x and P = 20 , then

2(x + b) = 20 ( divide both sides by 2 )

x + b = 10 ( subtract x from both sides )

b = 10 - x

Using Pythagoras' identity in the right triangle

x² + (10 - x)² = 8²

x² + 100 - 20x + x² = 64 ( subtract 64 from both sides )

2x² - 20x + 36 = 0 ( divide through by 2 )

x² - 10x + 18 = 0 ← as required

(b)

solve using the method of completing the square

x² - 10x + 18 = 0 ( subtract 18 from both sides )

x² - 10x = - 18

add ( half the coefficient of the x- term )² to both sides

x² + 2(- 5)x + 25 = - 18 + 25

(x - 5)² = 7 ( take square root of both sides )

x - 5 = ± [tex]\sqrt{7}[/tex] ( add 5 to both sides )

x = 5 ± [tex]\sqrt{7}[/tex]

Then

x = 5 - [tex]\sqrt{7}[/tex] ≈ 2.35 ( to 3 significant figures )

x = 5 + [tex]\sqrt{7}[/tex] ≈ 7.65 ( to 3 significant figures )

By using the formula of acceleration, it can be calculated that

Time taken by an earthworm  to accelerate from a speed of 0.01 m/s to 0.02 m/s over a distance of 0.9 m is 59.88 s

What is acceleration?

At first, it is important to know about velocity

Distance travelled by a body per unit interval of time in a particular direction is called Velocity

The rate of change of velocity is called acceleration.

Acceleration is a vector quantity as the direction is taken into account.

Initial velocity of earthworm (u)= 0.01 m/s

Final velocity of earthworm (v) = 0.02 m/s

Distance (s) = 0.9 m

We know,

[tex]v^2 = u^2 + 2as\\0.02^2 = 0.01^2 + 2 \times a \times 0.9\\0.02^2 - 0.01^2 = 1.8a\\(0.02 + 0.01)(0.02 - 0.01) = 1.8a\\a = \frac{0.03 \times 0.01}{1.8}\\a = 0.000167 \ ms^{-2}[/tex]

Again,

v = u + at

0.02 = 0.01 + 0.000167[tex]\times[/tex] t

t = [tex]\frac{0.02 - 0.01}{0.000167}[/tex]

t = 59.88 s

Time taken by an earthworm  to accelerate from a speed of 0.01 m/s to 0.02 m/s over a distance of 0.9 m is 59.88 s

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The sum of the first and second integer is 44.

Let the numbers be x, x+2 and x+4.

If the second is subtracted from twice the sum of the first and third,the result is 52 more than the first. This will be:

2(x + 2 + x + 4) - (x + 2) = x + 52

2(2x + 6) - (x + 2) = x + 52

4x + 12 - x - 2 = x + 52

Collect like terms

4x - x - x = 52 - 12 + 2

2x = 42

Divide

x = 42/2

x = 21

Second Integer = x + 2 = 21 + 2 = 23

Third integer = x + 4 = 21 + 4 = 25

Sum = 21 + 23 = 44

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1) The probability that a randomly chosen penny weighs less than 2.4 grams is; 0.00621.

2) The probability that the mean weight of 15 pennies is less than 2.4 grams is; 0

The central limit theorem in statistics tells us that the sampling distribution of the mean will usually be normally distributed, provided that the sample size is large enough.

We are given the following parameter;

Population mean; μ = 2.4 grams

Standard deviation; σ = 0.04 grams

1) The probability that a randomly chosen penny weighs less than 2.4 grams is: P(X < 2.4)

Formula for z-score is;

z = (X - μ)/σ

z = (2.4 - 2.5)/0.04

z = -2.5

The p-value from z-score calculator is; p = 0.00621.

2) The standard deviation for these 15 pennies will now be;

σ' = 0.04/√15

σ' = 0.0103

The probability that the mean weight of 15 pennies is less than 2.4 grams is expressed as;

P(X' < 2.4)

z = (2.4 - 2.5)/0.0103

z = -9.709

The p-value from z-score calculator is; p = 0

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Gabriel owns a stack of nickels which is 2 inches tall and thickness of each nickel is 0.08 inch then number of nickels with Gabriel is equal to 25.

As given in the question,

Gabriel has 2 inches tall stack of nickel

Thickness of each nickel = 0.08inch.

Let 'n' be the number of nickels Gabriel has .

Required equation to get the number of nickel is equal to :

0.08 × n = 2

⇒ n = 2 / 0.08

⇒ n = 200/ 8

⇒ n = 25

Therefore,  Gabriel owns a stack of nickels which is 2 inches tall and thickness of each nickel is 0.08 inch then number of nickels with Gabriel is equal to 25.

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Answer 0.04

Step-by-step explanation:

because 0.08 divided by two is 0.04

Answer:

96, $ 144

Step-by-step explanation:

Step-by-step explanation:

10 hours per week: $10.50 x 10 = $105.00

20 hours per week: $105.00 x 2 = $210.00

40 hours per week: $210.00 x 2 = $420.00

The cost of topsoil per ton would be $48.

The cost of pea gravel per ton is  $20.

The cost of mulch per ton is $479.

What is the linear equations in two variable?

A linear equation in two variables is one that is written in the form ax + by + c=0, where a, b, and c are real numbers and the coefficients of x and y, i.e. a and b, are not equal to zero. For example, 10x+4y = 3 and -x+5y = 2 are two-variable linear equation.

We have,

7 tons of topsoil, 6 tons of mulch, and 3 tons of pea gravel for $3270.

5 tons of topsoil, 5 tons of mulch, and 5 tons of pea gravel for $3015.

Let,

cost of topsoil per ton = x

cost of pea gravel per ton = x - 28

cost of mulch per ton = m

7x + 6m + 3(x - 28)  = 3270

7x + 6m + 3x - 84  = 3270

10x + 6m = 3270 + 84

10x + 6m = 3354 -------------(i)

5x + 5m + 5(x - 28)  = 3015

5x + 5m + 5(x - 28)  = 3015

10x + 5m = 3015 - 140

10x + 5m = 2875 -----------(II)

Subtracting equation (II) from (I), we get

   10x + 6m = 3354

-   10x + 5m = 2875

Solving we get

       m = 479

Put m = 479 in equation(I), we get

           10x + 6m = 3354

           10x + 6(479) = 3354

           10x = 480

              x = 480/10

              x = 48

Hence,

The cost of topsoil per ton would be = x = $48.

The cost of pea gravel per ton = x - 28 = 48 - 28 = $20.

The cost of mulch per ton = m = $479.

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Answer:

(B)={8,10,15,18,20}

Step-by-step explanation:

Answer:

see explanation

Step-by-step explanation:

this can be expressed in ratio form and fractional form , that is

45 : 30 ( divide each part by 15 )

= 3 : 2 ← in simplest form

= [tex]\frac{3}{2}[/tex]

Option (B) (3 + 2i) (3 - 2i) is the expression that has the whole number product.

Whole number:

Whole number refers the numbers without fractions and it is a collection of positive integers and zero and represented by the symbol “W” and the set of numbers are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,……………}.

Given,

Here we have the following expression:

(A) (4- 2i) (4 - 2i)

(B) (3 + 2i) (3 - 2i)

(C) (-6+ i)(3- i)

(D) (-4+3i)(4-3i)

Now, we need to find the whole number product.

In order to find the solution we have to perform the multiplication on the given expression,

First expression will gives the following,

(A) (4 - 2i) (4 - 2i)

When we apply the (a-b)² formula in it then we get,

=> 4²+ (2i)² - (2 x 4 x 2i)

=> 16 - 4 - 16i

=> 12 - 16i

Which is not a whole number product.

Second expression,

(B) (3 + 2i) (3 - 2i)

Apply the (a + b)(a - b) formula in it,

Then we get,

=> 3² - (2i)²

=> 9 + 4

=> 13

Therefore, this one is the whole number product.

Third expression,

(C) (-6 + i) (3 - i)

When we perform the multiplication on it, then we get,

=> -18 + 6i +3i -i²

=> -18 + 9i + 1

=> -17 + 9i

his one is not a whole number product.

Final expression,

(D) (-4 + 3i) (4 - 3i)

When we perform the multiplication on it, then we get,

=> -16 + 12i + 12i - (3i)²

=> -16 + 24i + 9

=> -7 + 24i

This one is also no the whole number product.

To know more about Whole number here.

https://brainly.com/question/19161857

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[tex]~\hfill \stackrel{\textit{\large distance between 2 points}}{d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2}}~\hfill~ \\\\[-0.35em] ~\dotfill\\\\ E(\stackrel{x_1}{2}~,~\stackrel{y_1}{3})\qquad F(\stackrel{x_2}{3}~,~\stackrel{y_2}{1}) ~\hfill EF=\sqrt{(~~ 3- 2~~)^2 + (~~ 1- 3~~)^2} \\\\\\ ~\hfill EF=\sqrt{( 1)^2 + ( -2)^2} \implies \boxed{EF=\sqrt{ 5 }}[/tex]

[tex]F(\stackrel{x_1}{3}~,~\stackrel{y_1}{1})\qquad D(\stackrel{x_2}{-1}~,~\stackrel{y_2}{-1}) ~\hfill FD=\sqrt{(~~ -1- 3~~)^2 + (~~ -1- 1~~)^2} \\\\\\ ~\hfill FD=\sqrt{( -4)^2 + ( -2)^2} \implies \boxed{FD=\sqrt{ 20 }} \\\\\\ D(\stackrel{x_1}{-1}~,~\stackrel{y_1}{-1})\qquad E(\stackrel{x_2}{2}~,~\stackrel{y_2}{3}) ~\hfill DE=\sqrt{(~~ 2- (-1)~~)^2 + (~~ 3- (-1)~~)^2} \\\\\\ ~\hfill DE=\sqrt{( 3)^2 + (4)^2} \implies DE=\sqrt{ 25 }\implies \boxed{DE=5} \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]

[tex]E(\stackrel{x_1}{2}~,~\stackrel{y_1}{3})\qquad F(\stackrel{x_2}{3}~,~\stackrel{y_2}{1}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{1}-\stackrel{y1}{3}}}{\underset{run} {\underset{x_2}{3}-\underset{x_1}{2}}} \implies \cfrac{ -2 }{ 1 } \implies - 2 \\\\[-0.35em] ~\dotfill[/tex]

[tex]F(\stackrel{x_1}{3}~,~\stackrel{y_1}{1})\qquad D(\stackrel{x_2}{-1}~,~\stackrel{y_2}{-1}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{-1}-\stackrel{y1}{1}}}{\underset{run} {\underset{x_2}{-1}-\underset{x_1}{3}}} \implies \cfrac{ -2 }{ -4 } \implies \cfrac{1 }{ 2 } \\\\[-0.35em] ~\dotfill[/tex]

[tex]D(\stackrel{x_1}{-1}~,~\stackrel{y_1}{-1})\qquad E(\stackrel{x_2}{2}~,~\stackrel{y_2}{3}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{3}-\stackrel{y1}{(-1)}}}{\underset{run} {\underset{x_2}{2}-\underset{x_1}{(-1)}}} \implies \cfrac{3 +1}{2 +1} \implies \cfrac{4 }{ 3 }[/tex]