Find the volume of the solid below.

In a normal distribution, approximately 89% of the observations are expected to be contained within 1.25 standard deviations of the mean.

This can be determined using the empirical rule, also known as the 68-95-99.7 rule, which states that:

- Approximately 68% of the observations fall within 1 standard deviation of the mean.

- Approximately 95% of the observations fall within 2 standard deviations of the mean.

- Approximately 99.7% of the observations fall within 3 standard deviations of the mean.

Since 1.25 standard deviations is between 1 and 2 standard deviations, we can estimate that about 89% of the observations will fall within this range.

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To achieve ten times higher accuracy in the calculation without changing the upper bound value obtained in part (a), we can adjust the stopping criterion or convergence condition for the iterative methods used.

For the Secant method, we can modify the convergence condition to stop the iteration when the absolute difference between consecutive approximations, |p_n - p_(n-1)|, becomes smaller than the desired tolerance. By decreasing the tolerance by a factor of ten, we can achieve higher accuracy while keeping the same upper bound value.

Similarly, for the Method of False Position, we can modify the convergence condition to stop the iteration when the absolute difference between the current approximation p_n and the previous approximation p_(n-1), |p_n - p_(n-1)|, becomes smaller than the desired tolerance. By decreasing the tolerance by a factor of ten, we can obtain a more accurate result without changing the upper bound value calculated in part (a).

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The period of the function is 20 minutes, and the coefficient in front of x is (2π/20).

We can utilize the cosine function, which repeats in a circular manner, to simulate the scenario with a periodic function.

Let's do a detailed analysis of the issue.

About 2,150 miles separate Atlanta from Seattle on this particular flight.

This indicates that the plane departs from Atlanta at a distance of 0 miles and travels 2,150 miles to arrive in Seattle (x = 2,150).

After flying 30 miles past Seattle, the aircraft begins a circling holding pattern.

The revised distance from Atlanta is 2,150 + 30 = 2,180 miles after the jet flies an extra 30 miles beyond Seattle.

Every 20 minutes, the aircraft makes one full round.

This indicates that the function will last for 20 minutes.

Let's first create the cosine function's equation: y = Acos(Bx).

The coefficients A and B determine the frequency (number of cycles) and amplitude (highest value) of the function, respectively.

Now, we can determine the values of A and B.

The difference between the maximum and smallest values of y is equal to half of the cosine function's amplitude (A).

When the jet is in the circular holding pattern, the distance from Atlanta can be as far as 2,180 miles, and it can be as close as 0 miles when it is at Atlanta.

The amplitude A is therefore (1,090 miles) = (2,180 - 0) / 2.

The frequency (B) of the cosine function is determined by the formula: B = 2π / T, where T is the period. In this case, T = 20 minutes, so B = 2π / 20.

Therefore, the equation for the periodic function that models this situation is:

y = 1090cos((2π/20)x)

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(a) To find the marginal cost function C'(x), we need to take the derivative of the cost function C(x) with respect to x.

To determine how fast the cost is going up at a production level of 100 teddy bears, we substitute x = 100 into the marginal cost function:

C'(100) = (38 - 0.16)(100)

           = 38 - 16

           = $22 per teddy bear.

The exact cost of producing the 101st teddy bear can be found by substituting x = 101 into the cost function:

C(101) = 100 + 38(101) - 0.08(101)^2  

         = [tex](100 + 38(101) - 0.08(101)^{2} )[/tex]

         = = $434.92

Thus, there is a difference of $434.92 - $434 = $0.92.

(b) The average cost function C(x) is given by:

C(x) =  [tex]\frac{C(x)}{x}[/tex]  =  [tex]\frac{100 + 38x - (0.08)^{2} }{x}[/tex]

To evaluate C(100), we substitute x = 100 into the average cost function:

C(100) =   [tex]\frac{100 + 38(100) - (0.08)(100)^{2} }{100}[/tex]

          = $138 per teddy bear.

The answer tells us that the average cost of producing the first hundred teddy bears is $138 per teddy bear.

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Based on the given information, the search boat is positioned 65 meters due east of the shore, and the angle of depression between the shore and the hypothetical location of The Howler and its treasure is 30°.

To find the straight-line distance between the treasure and the shore, we can use trigonometric functions to calculate the length of the hypotenuse of the right triangle formed by the shore, the search boat, and the treasure. Let's denote the length of the straight-line distance between the treasure and the shore as d. In the right triangle formed by the shore, the search boat, and the treasure, the side opposite the 30° angle is d (the distance between the treasure and the shore), and the side adjacent to the 30° angle is 65 meters (the distance between the search boat and the shore).

Using the trigonometric function tangent (tan), we can set up the equation:

tan(30°) = opposite/adjacent

tan(30°) = d/65

To find the value of d, we rearrange the equation:

d = 65 * tan(30°)

d ≈ 65 * 0.577

d ≈ 37.51

Therefore, the straight-line distance between the treasure and the shore is approximately 37.51 meters.

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Based on the given information, the search boat is positioned 65 meters due east of the shore, and the angle of depression between the shore and the hypothetical location of The Howler and its treasure is 30°.

To find the straight-line distance between the treasure and the shore, we can use trigonometric functions to calculate the length of the hypotenuse of the right triangle formed by the shore, the search boat, and the treasure. Let's denote the length of the straight-line distance between the treasure and the shore as d. In the right triangle formed by the shore, the search boat, and the treasure, the side opposite the 30° angle is d (the distance between the treasure and the shore), and the side adjacent to the 30° angle is 65 meters (the distance between the search boat and the shore).

Using the trigonometric function tangent (tan), we can set up the equation:

tan(30°) = opposite/adjacent

tan(30°) = d/65

To find the value of d, we rearrange the equation:

d = 65 * tan(30°)

d ≈ 65 * 0.577

d ≈ 37.51

Therefore, the straight-line distance between the treasure and the shore is approximately 37.51 meters.

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a. The original quantified statement can be expressed in an equivalent way as follows: "For every element x in a particular set, there exists a property P(x) that holds true." This means that each element in the set possesses the specific property P(x).

b. The negation of the quantified statement would be: "There exists an element x in the particular set such that the property P(x) does not hold true." In this case, the negation asserts that at least one element in the set does not possess the property P(x).

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A rational zero of the function f(x) = 2x^3 + 15x^2 - 4x + 32 is x = -4/2.

The rational zero theorem states that if a polynomial function has a rational root (zero), it can be expressed as p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

In this case, the constant term is 32 and the leading coefficient is 2. Factors of 32 are ±1, ±2, ±4, ±8, ±16, ±32, and factors of 2 are ±1, ±2. By testing the possible combinations, we find that -4/2 is a rational zero.

This means that when x = -4/2, the polynomial function will equal zero.

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The entry that should be made by Mifflin Company on the maturity date is as follows: Debit: Notes Receivable $7,400

Credit: Accounts Receivable - Payton Summers $7,400

This entry records the collection of the note receivable from Payton Summers, replacing the accounts receivable. The debit to Notes Receivable reduces the balance in the Notes Receivable account, while the credit to Accounts Receivable - Payton Summers reduces the outstanding balance in the accounts receivable from the customer.

It's important to note that the entry assumes that the note is paid in full on the maturity date. If there were any adjustments or additional entries required (e.g., interest accrual), they would need to be considered and recorded separately

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To determine which order cycle length would require the buyer to hold the most total inventory during lead time, we need to consider the combination of order cycle length and the variability in lead time.

In this case, the order cycle length refers to the time between placing an order and receiving the inventory, and the "+/-" represents the variability or uncertainty in the lead time.

To calculate the total inventory held during lead time, we need to consider the maximum lead time within the given range (positive or negative) and add it to the order cycle length.

Let's calculate the total inventory for each option:

a. 10 days order cycle length, +/- 4 days lead time variability: Total inventory = 10 + 4 = 14 days

b. 10 days order cycle length, +/- 2 days lead time variability: Total inventory = 10 + 2 = 12 days

c. 9 days order cycle length, +/- 3 days lead time variability: Total inventory = 9 + 3 = 12 days

d. 8 days order cycle length, +/- 5 days lead time variability: Total inventory = 8 + 5 = 13 days

Comparing the total inventory calculations, option a has the highest total inventory requirement during lead time with 14 days. Therefore,  a. 10 days, +/4 days would require the buyer to hold the most total inventory during lead time.

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The multiplication rule of indices is showcased in the picture where the power of multiplied values with the same base are added.

In indices, when two or more values having the same base are multiplied, the value of it's power are added to give a singular value with the same base and the power being the sum of the power values.

[tex] {a }^{m } \times {a}^{n} = {a}^{m + n} [/tex]

The multiplication rule of indices is an established rule and cannot be disproved once all conditions are met.

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Answer

To answer this problem you have to minus 8 from the quotient of 2 and r.  So you divide 2 and r and minus 8

The wheel shop has 43 bicycles, 40 tricycles, and 32 tandem bicycles in total.

Let's assume the number of bicycles in the shop is "b," the number of tricycles is "t," and the number of tandem bicycles is "d."

Based on the given information, the number of seats can be expressed as: 1b + 1t + 2d = 135. Similarly, the number of front handlebars can be expressed as: 1b + 1t + 1d = 118. Additionally, the number of wheels can be expressed as: 2b + 3t + 2d = 269.

We can solve this system of equations to find the values of b, t, and d. However, instead of providing the detailed calculations, we can solve the system using an algebraic tool.

Solving the system of equations, we find that there are 43 bicycles, 40 tricycles, and 32 tandem bicycles in the wheel shop.

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Since φ satisfies all the conditions of the First Isomorphism Theorem, we conclude that K/(K ∩ N) is isomorphic to (K N)/N.

What is First Isomorphism Theorem?

The First Isomorphism Theorem is a fundamental result in group theory that establishes a connection between a group homomorphism and the structure of the groups involved. It states that if φ: G → H is a group homomorphism with kernel K, then the quotient group G/K is isomorphic to the image of φ, denoted as φ(G). In other words, the cosets of the kernel K in G form a group isomorphic to the image of G under the homomorphism φ

To prove that K/(K ∩ N) is isomorphic to (K N)/N, where K is a subgroup of G and N is a normal subgroup of G, we can use the First Isomorphism Theorem. The theorem states that if φ: G → H is a homomorphism with kernel K, then G/K is isomorphic to φ(G).

Let's define a homomorphism φ: K → (K N)/N, where φ(k) = kN. We need to show that φ is well-defined, injective, surjective, and preserves the group operation.

Well-defined: We need to show that if k1, k2 ∈ K and k1N = k2N, then φ(k1) = φ(k2). Since k1N = k2N, it implies that k1⁻¹k2 ∈ N. Since N is a normal subgroup of G and K is a subgroup of G, it follows that (k1⁻¹k2)k ∈ K for any k ∈ K. Hence, φ is well-defined.

Injective: We need to show that if φ(k1) = φ(k2), then k1 = k2. If φ(k1) = φ(k2), it implies that k1N = k2N, which means k1⁻¹k2 ∈ N. Since N is a subgroup of G, k1⁻¹k2 ∈ N implies k1⁻¹k2N = N. This implies k1⁻¹k2 ∈ K ∩ N. As K ∩ N contains only the identity element (since N is a normal subgroup and K is a subgroup), we have k1⁻¹k2 = e (identity element), which gives k1 = k2. Hence, φ is injective.

Surjective: We need to show that for every coset aN in (K N)/N, there exists an element k ∈ K such that φ(k) = aN. Since aN is a coset in (K N)/N, we can write aN = knN for some k ∈ K and n ∈ N. Hence, φ(k) = knN = aN. Therefore, φ is surjective.

Group operation preservation: We need to show that φ preserves the group operation. Let k1, k2 ∈ K. Then φ(k1k2) = (k1k2)N = (k1N)(k2N) = φ(k1)φ(k2). Hence, φ preserves the group operation.

Since φ satisfies all the conditions of the First Isomorphism Theorem, we conclude that K/(K ∩ N) is isomorphic to (K N)/N.

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Answer:

Step-by-step explanation:

The root is the fractional part of an exponent and the power is the upper part of the exponent fraction

1)    [tex]\sqrt{x^{3} } = x^{\frac{3}{2} }[/tex]

2)    [tex]17^{\frac{1}{5} } =\sqrt[5]{17}[/tex]

3)   [tex]\sqrt[5]{y^{3} } = y^{\frac{3}{5} }[/tex]

4)   [tex]z^{\frac{2}{3} } =\sqrt[3]{z^{2} }[/tex]

The area bounded above by f(x) = -x^2 - 10x - 16 and bounded below by the x-axis over the interval [-8, -2] is 1208/3 square units.

To determine the area bounded above by f(x) = -x^2 - 10x - 16 and bounded below by the x-axis over the interval [-8, -2], we need to find the definite integral of the absolute value of the function f(x) - g(x) over the given interval.

The absolute value of f(x) - g(x) is |(-x^2 - 10x - 16) - (2x + 16)| = |-x^2 - 12x - 32|. We need to find the integral of this absolute value function from x = -8 to x = -2.

∫[-8,-2] |-x^2 - 12x - 32| dx

To solve this integral, we need to break it up into two separate integrals based on the sign of the function.

For -8 ≤ x ≤ -4, the expression inside the absolute value becomes positive:

∫[-8,-4] (-x^2 - 12x - 32) dx

For -4 ≤ x ≤ -2, the expression inside the absolute value becomes negative:

∫[-4,-2] (x^2 + 12x + 32) dx

Evaluating the integrals separately, we get:

∫[-8,-4] (-x^2 - 12x - 32) dx = [(1/3)x^3 + 6x^2 + 32x] [-8,-4]

= [(-64/3) + 96 - 256] - [(64/3) + 96 + 128]

= -160 - (352/3)

= -480/3 - 352/3

= -832/3

∫[-4,-2] (x^2 + 12x + 32) dx = [(1/3)x^3 + 6x^2 + 32x] [-4,-2]

= [(-32/3) + 48 - 128] - [(-8/3) + 24 + 64]

= -112 - (40/3)

= -336/3 - 40/3

= -376/3

Now, to find the area, we take the absolute value of the sum of these two integrals:

Area = |(-832/3) + (-376/3)|

= |(-832 - 376)/3|

= |(-1208)/3|

= 1208/3

Therefore, the area bounded above by f(x) = -x^2 - 10x - 16 and bounded below by the x-axis over the interval [-8, -2] is 1208/3 square units.

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Answer: Not sure what you mean but

Step-by-step explanation:

Rashad can let his friends know he is ok by sending them a message or snap letting them know he is ok. He can also call or text them to reassure them that is alright.

(a) The largest possible value of Lz in units of ℏ for states with l=3 is 3ℏ.

(b) The value of L in units of ℏ for states with l=3 is 3ℏ. The maximum possible Lz is equal to L, so they are equal.

(c) In a classical vector model, for each allowed value of Lz, the vector makes an angle with the axis that depends on the specific value of Lz and the orientation of the vector. Without further information, it is not possible to determine the exact angle.

In quantum mechanics, the angular momentum operator Lz measures the projection of the angular momentum along the z-axis. For states with l=3, the maximum possible value of Lz is equal to l, which is 3.

The total angular momentum L for states with l=3 is also equal to l, which is 3. In this case, the maximum possible value of Lz is equal to L. Therefore, L and the maximum possible Lz are equal.

In a classical vector model, the orientation of the vector is determined by the values of Lx, Ly, and Lz. The angle that the vector makes with the axis depends on the specific values of Lx, Ly, and Lz. Without knowing these values, it is not possible to determine the exact angle.

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To help you identify which correlation goes with which scatterplot, here's a brief explanation of the correlation coefficients provided:

1. 0.62: This positive correlation indicates a moderate, positive relationship between the two variables. As one variable increases, the other also tends to increase. In the scatterplot, you'll see a rough upward trend in the data points, but they might not be tightly clustered around a line.
2. -0.93: This strong negative correlation implies a significant, negative relationship between the two variables. As one variable increases, the other tends to decrease. In the scatterplot, you'll see a clear downward trend in the data points, closely clustered around a line.
3. -0.02: This near-zero correlation suggests that there is virtually no relationship between the two variables. The scatterplot will show a random distribution of data points without any apparent pattern.
To determine which correlation goes with which scatterplot, examine the scatterplots closely and identify the trends described above. Match each scatterplot to the corresponding correlation based on the strength and direction of the relationship between the variables.

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The length of the hypotenuse for the right angled triangle given is 21.4.

Given a right angled triangle.

We have to find the length of the hypotenuse.

We know by Pythagoras theorem, square of the hypotenuse is equal to the sum of the squares of the legs.

Using this,

Hypotenuse² = (leg 1)² + (leg 2)²

(JL)² = (JK)² + (KL)²

       = 13² + 17²

       = 458

JL = √458 = 21.4

Hence the length is 21.4.

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The inequality of the statement The distance, d, to the nearest exit is no less than 30 meters is d ≥ 30

From the question, we have the following parameters that can be used in our computation:

The distance, d, to the nearest exit is no less than 30 meters

Represent the distance with d

So, we have

d is no less than 30 meters

In inequality, no less than means greater than or equal to

So, we have

d ≥ 30

Hence, the inequality of the statement is d ≥ 30

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To show the formation of a sulfide ion from a neutral sulfur atom, we need to add two electrons to the sulfur atom, as sulfide ion has a charge of -2. Therefore, the equation for this process is: S + 2e- → S2-

In this equation, S represents the neutral sulfur atom, while S2- represents the sulfide ion that is formed after the addition of two electrons. This reaction is a reduction reaction, as sulfur is gaining two electrons to form a negatively charged ion.
In summary, the equation S + 2e- → S2- shows the formation of the sulfide ion from a neutral sulfur atom by adding two electrons to it. This equation highlights the importance of electron transfer in chemical reactions and how it can lead to the formation of new compounds.

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The solution to the proportional relationship in this problem is given as follows:

x = -5.

The solution is not extraneous, as x = -5 does not make the denominator of any of the fractions zero.

A proportional relationship is a type of relationship between two quantities in which they maintain a constant ratio to each other.

The constant ratio in the context of this problem is given as follows:

4/(x - 1) = 2/(x + 2)

Applying cross multiplication, we can obtain the value of x as follows:

4(x + 2) = 2(x - 1)

4x + 8 = 2x - 2

2x = -10

x = -5.

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The linear regression model for the weekly cost data is given by C(z) = ma + k, where z is the independent variable representing the weekly demand.



The values of m and k are determined through the regression analysis. Using this model, we can estimate the total weekly cost for a given weekly demand and calculate the per unit variable cost and weekly fixed cost.To find the linear regression model, we need to perform a regression analysis on the given weekly cost data. This analysis will determine the values of m and k in the equation C(z) = ma + k, where C(z) represents the weekly cost and z represents the weekly demand.

Once the values of m and k are obtained, we can use the model to estimate the total weekly cost when the weekly demand is 210. By plugging in z = 210 into the equation C(z) = ma + k, we can calculate the total weekly cost rounded to the nearest dollar.The per unit variable cost can be determined by the value of m in the model. It represents the change in cost per unit change in demand. By rounding m to one decimal place, we can obtain the per unit variable cost rounded to one decimal place.

The weekly fixed cost can be determined by the value of k in the model. It represents the cost that does not depend on the weekly demand. By rounding k to the nearest integer, we can obtain the weekly fixed cost rounded to the nearest dollar.Overall, by applying the linear regression model to the given data, we can estimate the total weekly cost, calculate the per unit variable cost, and determine the weekly fixed cost of producing sleeping bags.

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Here the correct answer is (a) 0.05 .The probability of the random variable x, which is uniformly distributed between 70 and 90, having a value between 80 and 95 can be determined by calculating the area under the probability density function (PDF) curve within that range.

In the given scenario, x follows a uniform distribution with a minimum value of 70 and a maximum value of 90. Since the distribution is uniform, the PDF is constant within the interval [70, 90] and zero outside that range. To find the probability of x lying between 80 and 95, we need to calculate the proportion of the total area under the PDF curve within that range.

The range of 80 to 95 is partially outside the interval [70, 90], extending beyond the maximum value of 90. Therefore, the probability of x falling within this range is zero, as there is no overlap between the defined range of x and the desired range of 80 to 95. Hence, the correct answer is (a) 0.05, indicating that the probability is negligible or non-existent in this case.

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hello

the answer to the question is:

(a) y = 65°

(b) z = 110°

(c) z = 25°

(d) y = 125°

(e) z = 80°

The probability of drawing four cards of four different suits from a standard deck of cards is approximately 0.588.

To calculate the probability, we first determine the total number of possible outcomes. There are 52 cards in a deck, and we are drawing four cards without replacement, so the total number of possible outcomes is given by the combination formula C(52, 4) = 270,725.

Next, we calculate the number of favorable outcomes, which is the number of ways to choose one card from each suit. For the first card, we have 52 options. For the second card, there are 39 remaining cards of different suits. Similarly, for the third and fourth cards, we have 26 and 13 options, respectively. Therefore, the number of favorable outcomes is 52 * 39 * 26 * 13 = 1,690,728.

Finally, we divide the number of favorable outcomes by the total number of possible outcomes to obtain the probability: 1,690,728 / 270,725 ≈ 0.588.

Therefore, the probability of drawing four cards of four different suits from a well-shuffled deck is approximately 0.588, or 58.8%.

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Answer:

Step-by-step explanation:

You want the inequalities and graph representing the given scenario regarding a boy's buying plans for apples and mangoes.

The problem statement tells us to use 'a' to represent the number of apples, and 'm' to represent the number of mangoes the boy buys. His constraints are ...

  a ≥ 5 . . . . . . . . he buys at least 5 apples

  m ≥ 5 . . . . . . . he buys at least 5 mangoes

  5a +3m ≤ 75 . . . . he spends at most 75

  5a -3m ≥ 5 . . . . . he spends at least 5 more on apples

Using x and y for 'a' and 'm', the graph is attached. The corners of the feasible region have their vertices identified. The feasible region is the area overlaid by 4 shadings.

(i) The maximum number of apples he can buy is 12 (lower right corner)

(ii) The maximum number of mangoes he can buy is 11 (nearest integer to the top vertex)

__

Additional comment

When there are this many inequalities, it sometimes works well to reverse them when graphing. That way, the feasible region is white, and the non-feasible areas are shaded.

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From the addition rule of integral, the evaluate value of the definite integral,[tex]\int_{0}^{\frac{\pi }{3}} (sec²x + 3x )dx [/tex], is equals to the [tex] \sqrt{3} + \frac{π²}{6}[/tex].

Definite integral of f(x) is a number and represents the area under the curve of a function f(x) from x=a to x= b.

We have an definite integral, [tex]\int_{0}^{\frac{\pi }{3}} (sec²x + 3x )dx [/tex]. We have to evaluate it's value. Using the addition rule of integral, [tex]\int_{0}^{\frac{\pi }{3}}(sec²x + 3x )dx = \int_{0}^{\frac{π}{3}} sec ²x dx + \int_{0}^{\frac{π}{3}} 3xdx [/tex].

Apply the general integral rules and the fundamental theorem of integrals,

[tex] = [tan(x)]_{0}^{\frac{π}{3} }+ 3\int_{0}^{\frac{π}{3}}xdx ( using the trigonometric rule in indefinite integral, [tex] \int sec² u du = [tan(u) + C] [/tex])

[tex] = [tan(\frac{π}{3}) - tan(0) ]+ 3 [\frac{x²}{2}]_{0}^{\frac{π}{3}}[/tex] ( from the indefinite integral using the expontent rule, [tex] \int u^{n }du = \frac{u^{n + 1}}{n + 1} + C] [/tex])

[tex] = \sqrt{3} + \frac{3}{2}(\frac{π}{3})²[/tex]

[tex] = \sqrt{3} + \frac{π²}{6}[/tex].

Hence, required value is [tex] \sqrt{3} + \frac{π²}{6}[/tex].

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Complete question:

Evaluate the definite intergral integral from [tex]\int_{0}^{\frac{\pi }{3}} (sec²x + 3x )dx [/tex].

If the slopes are equal, we can conclude that the segment joining the midpoints of two sides of a triangle is parallel to the third side.

To prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side, we can follow these steps:

Assign (x, y) coordinates to points A, B, and C: Let's assume that point A has coordinates (x1, y1), point B has coordinates (x2, y2), and point C has coordinates (x3, y3).

Calculate the coordinates of the midpoints: The midpoint of AB, denoted as M, can be calculated as ((x1 + x2)/2, (y1 + y2)/2), and the midpoint of AC, denoted as N, can be calculated as ((x1 + x3)/2, (y1 + y3)/2).

Calculate the slope of MN: The slope of a line passing through two points (x1, y1) and (x2, y2) is given by (y2 - y1)/(x2 - x1). So, the slope of MN is ((y1 + y3)/2 - y1)/((x1 + x3)/2 - x1).

Calculate the slope of AB: Similarly, the slope of AB is (y2 - y1)/(x2 - x1).

Show that the slopes are equal: Compare the slope of MN with the slope of AB. Simplify the expressions and check if they are equal. If the slopes are equal, it means that the segment joining the midpoints is parallel to the third side of the triangle.

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