answer：Okay, so I have this problem here: I need to find the sum of the coordinates of the midpoint of a segment with endpoints at (8, -4) and (-2, 10). Then, I also need to find the distance from this midpoint to one of the endpoints, specifically (8, -4). Alright, let's break this down step by step.First, I remember that the midpoint of a segment can be found by averaging the x-coordinates and the y-coordinates of the endpoints. So, the formula for the midpoint (M) is:M = ((x₁ + x₂)/2, (y₁ + y₂)/2)Let me plug in the given points into this formula. The first endpoint is (8, -4), so x₁ is 8 and y₁ is -4. The second endpoint is (-2, 10), so x₂ is -2 and y₂ is 10.Calculating the x-coordinate of the midpoint:(8 + (-2))/2 = (8 - 2)/2 = 6/2 = 3Calculating the y-coordinate of the midpoint:(-4 + 10)/2 = (6)/2 = 3So, the midpoint is (3, 3). Now, the problem asks for the sum of the coordinates of this midpoint. That means I need to add the x and y values together:3 + 3 = 6Alright, so the sum of the coordinates is 6. That part seems straightforward.Next, I need to find the distance from this midpoint (3, 3) to the endpoint (8, -4). I recall that the distance between two points (x₁, y₁) and (x₂, y₂) is given by the distance formula:Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]Let me apply this formula to the points (3, 3) and (8, -4). Here, (x₁, y₁) is (3, 3) and (x₂, y₂) is (8, -4).Calculating the differences in x and y:x₂ - x₁ = 8 - 3 = 5y₂ - y₁ = -4 - 3 = -7Now, squaring these differences:5² = 25(-7)² = 49Adding these squared differences:25 + 49 = 74Taking the square root of this sum to find the distance:√74Hmm, √74 is an irrational number, so I can't simplify it further. It looks like the distance is √74 units.Let me just double-check my calculations to make sure I didn't make any mistakes. For the midpoint, I added 8 and -2 to get 6, divided by 2 to get 3 for the x-coordinate. For the y-coordinate, I added -4 and 10 to get 6, divided by 2 to get 3. That seems correct.For the distance, I subtracted 3 from 8 to get 5, and 3 from -4 to get -7. Squaring those gives 25 and 49, which add up to 74. The square root of 74 is indeed √74. Okay, that checks out.I think I've got it right. The sum of the coordinates of the midpoint is 6, and the distance from the midpoint to the endpoint (8, -4) is √74.

answer：Okay, so I have this problem here: "In a throw of a coin, the probability of getting a certain outcome is 0.5. What is the outcome?" Hmm, let me try to figure this out step by step.First, I know that when you throw a coin, there are two possible outcomes: heads or tails. That's pretty basic. So, if I think about probability, it's usually the number of favorable outcomes divided by the total number of possible outcomes. In this case, if I want to find the probability of getting a certain outcome, I need to consider how many ways that outcome can happen compared to all possible outcomes.Since there are only two outcomes—heads or tails—the total number of possible outcomes is 2. Now, if the probability of a certain outcome is 0.5, that means the number of favorable outcomes divided by 2 equals 0.5. Let me write that down:Probability = Number of favorable outcomes / Total number of possible outcomesSo,0.5 = Number of favorable outcomes / 2To find the number of favorable outcomes, I can rearrange the equation:Number of favorable outcomes = 0.5 * 2 = 1Okay, so there's 1 favorable outcome. Since there are two possible outcomes (heads or tails), the favorable outcome must be either heads or tails. But the problem doesn't specify which one, so I guess it could be either.Wait, but the question is asking for the outcome, not the probability. So, if the probability is 0.5, the outcome is either heads or tails. But I'm not sure if I'm supposed to specify which one or just recognize that it's one of them.Let me think again. In a fair coin toss, the probability of getting heads is 0.5, and the probability of getting tails is also 0.5. So, if the probability is 0.5, the outcome is either heads or tails. But since the question is phrased as "the probability of getting a certain outcome is 0.5," it's likely referring to one specific outcome, not both.But wait, in reality, both outcomes have the same probability. So, maybe the question is trying to get me to recognize that both outcomes have a probability of 0.5, but it's phrased in a way that makes it seem like there's a single outcome with that probability.Hmm, I'm a bit confused now. Let me try to clarify. If I have a coin, and I toss it, there are two possible outcomes: heads or tails. Each has an equal chance of occurring, so each has a probability of 0.5. So, if the question is asking for the outcome that has a probability of 0.5, it's either heads or tails.But the way the question is phrased, it seems like it's expecting a specific answer, like "heads" or "tails." But since both have the same probability, I'm not sure which one to choose. Maybe the question is trying to test my understanding that in a fair coin toss, each outcome has a 0.5 probability.Alternatively, maybe the question is trying to trick me into thinking that there's only one outcome with a probability of 0.5, but in reality, both have that probability. So, perhaps the answer is that both outcomes have a probability of 0.5.Wait, but the question specifically says "the probability of getting a certain outcome is 0.5." So, it's referring to a single outcome. Therefore, the outcome is either heads or tails, each with a probability of 0.5.But I'm still not entirely sure. Maybe I should consider that in some cases, people might refer to the combined probability of getting either heads or tails, but that would be 1, not 0.5. So, no, that doesn't make sense.I think the safest answer is that the outcome is either heads or tails, each with a probability of 0.5. But since the question is asking for "the outcome," maybe it's expecting me to recognize that both outcomes have that probability, but it's phrased in a way that makes it seem like there's a single answer.Alternatively, maybe the question is trying to get me to think about the fact that in a fair coin toss, each outcome has a probability of 0.5, so the outcome is either heads or tails.I'm going to go with that. The outcome is either heads or tails, each with a probability of 0.5.

answer：Okay, so I have this problem here about an arithmetic sequence. Let me try to figure it out step by step. First, the problem says that Sₙ is the sum of the first n terms of an arithmetic sequence {aₙ}. It also gives me that a₂ + a₅ + a₈ = 15, and I need to find S₉. The options are A: 15, B: 30, C: 45, D: 60.Alright, let's recall what an arithmetic sequence is. It's a sequence where each term increases by a constant difference. So, if I denote the first term as a₁ and the common difference as d, then the nth term aₙ can be expressed as a₁ + (n-1)d.Given that, let's write down the expressions for a₂, a₅, and a₈.- a₂ = a₁ + d- a₅ = a₁ + 4d- a₈ = a₁ + 7dSo, the sum a₂ + a₅ + a₈ would be:(a₁ + d) + (a₁ + 4d) + (a₁ + 7d) = 3a₁ + 12dAnd this sum is equal to 15, as given. So:3a₁ + 12d = 15Hmm, maybe I can simplify this equation. Let's divide both sides by 3:a₁ + 4d = 5Wait, that's interesting because a₁ + 4d is actually a₅. So, a₅ = 5.That's a useful piece of information. Now, I need to find S₉, which is the sum of the first 9 terms of this arithmetic sequence.I remember that the sum of the first n terms of an arithmetic sequence is given by:Sₙ = n/2 * (2a₁ + (n - 1)d)Alternatively, it can also be written as:Sₙ = n/2 * (a₁ + aₙ)Since I know a₅, maybe I can use that to find the sum. Let me think.In an arithmetic sequence, the average of the terms equidistant from the ends is equal to the middle term. Since 9 is an odd number, the middle term is the 5th term, which is a₅.So, the average of the first 9 terms is a₅, which is 5. Therefore, the sum S₉ is 9 times the average, which is 9 * 5 = 45.Wait, that seems straightforward. Let me verify it using the sum formula to make sure I didn't make a mistake.Using the formula Sₙ = n/2 * (a₁ + aₙ), I need to find a₉. Since a₉ = a₁ + 8d.But I already know that a₅ = a₁ + 4d = 5. So, a₁ = 5 - 4d.Therefore, a₉ = (5 - 4d) + 8d = 5 + 4d.So, S₉ = 9/2 * (a₁ + a₉) = 9/2 * (5 - 4d + 5 + 4d) = 9/2 * 10 = 9 * 5 = 45.Okay, that confirms it. So, S₉ is indeed 45. Therefore, the correct answer should be C: 45.I think that makes sense. I used the property of arithmetic sequences where the middle term is the average, which simplifies the calculation a lot. It's always good to remember these properties because they can save a lot of time.Just to recap:1. Expressed a₂, a₅, a₈ in terms of a₁ and d.2. Summed them up and found that a₁ + 4d = 5, which is a₅.3. Recognized that in an arithmetic sequence with an odd number of terms, the middle term is the average.4. Calculated S₉ as 9 times the middle term, which is 45.5. Verified using the sum formula to ensure the answer is correct.So, I'm confident that the answer is 45, which is option C.**Final Answer**boxed{C}

answer：First, I need to calculate the total cost of the pencils. There are 75 pencils, and each pencil costs 2.00. So, I'll multiply the number of pencils by the cost per pencil to find the total cost for pencils.Next, I'll calculate the total cost of the pens. There are 30 pens, and each pen costs 14.00. I'll multiply the number of pens by the cost per pen to determine the total cost for pens.Finally, I'll add the total cost of pencils and the total cost of pens together to find the overall amount spent on both pens and pencils.

answer：Okay, so I have this trigonometry problem here. It says that sin(π/7 - x) equals -2/3, and I need to find the value of cos(9π/14 - x). The options are A) 2/3, B) -2/3, C) sqrt(5)/3, and D) -sqrt(5)/3. Hmm, let me think about how to approach this.First, I remember that there are some trigonometric identities that relate sine and cosine functions, especially cofunction identities. I think one of them is that cos(π/2 - θ) equals sinθ. But in this case, the angle inside the cosine function is 9π/14 - x, which doesn't immediately look like π/2 minus something. Maybe I can manipulate it to fit a known identity.Let me write down what I know:sin(π/7 - x) = -2/3I need to find cos(9π/14 - x). Hmm, maybe I can express 9π/14 in terms of π/2 or π/7. Let me calculate 9π/14. Since π is approximately 3.14, π/2 is about 1.57, and π/7 is roughly 0.448. So 9π/14 is about 2.01, which is a bit more than π/2. Wait, actually, 9π/14 is equal to π/2 + π/7 because π/2 is 7π/14, so 7π/14 + 2π/14 is 9π/14. Oh, that's a useful observation!So, 9π/14 is π/2 + π/7. That means cos(9π/14 - x) can be rewritten as cos(π/2 + π/7 - x). Let me write that down:cos(9π/14 - x) = cos(π/2 + π/7 - x)Now, I recall that there's an identity for cos(A + B). The formula is cosA cosB - sinA sinB. But in this case, A is π/2 and B is (π/7 - x). So, applying the identity:cos(π/2 + (π/7 - x)) = cos(π/2)cos(π/7 - x) - sin(π/2)sin(π/7 - x)I know that cos(π/2) is 0 and sin(π/2) is 1. So substituting these values in:= 0 * cos(π/7 - x) - 1 * sin(π/7 - x)= -sin(π/7 - x)But from the problem, sin(π/7 - x) is given as -2/3. So substituting that in:= -(-2/3)= 2/3Wait, so cos(9π/14 - x) equals 2/3? Let me double-check my steps to make sure I didn't make a mistake.1. I recognized that 9π/14 is π/2 + π/7, which seems correct because π/2 is 7π/14, and adding 2π/14 gives 9π/14.2. Then I used the cosine addition formula: cos(A + B) = cosA cosB - sinA sinB. That's a standard identity, so that should be right.3. Plugging in A = π/2 and B = π/7 - x, I correctly substituted cos(π/2) as 0 and sin(π/2) as 1.4. That simplified the expression to -sin(π/7 - x), which is correct.5. Given that sin(π/7 - x) is -2/3, substituting that in gives -(-2/3) = 2/3.Everything seems to check out. So, the value of cos(9π/14 - x) is 2/3, which corresponds to option A.But just to be thorough, let me consider if there's another way to approach this problem, maybe using the sine and cosine relationship directly. I know that cos(θ) = sin(π/2 - θ). So, can I express cos(9π/14 - x) in terms of sine?Let me try that:cos(9π/14 - x) = sin(π/2 - (9π/14 - x))Simplify the argument inside the sine function:π/2 is 7π/14, so:= sin(7π/14 - 9π/14 + x)= sin(-2π/14 + x)= sin(x - π/7)Hmm, so cos(9π/14 - x) equals sin(x - π/7). But from the problem, we know sin(π/7 - x) = -2/3. Notice that sin(x - π/7) is the same as -sin(π/7 - x) because sine is an odd function. So:sin(x - π/7) = -sin(π/7 - x) = -(-2/3) = 2/3Therefore, cos(9π/14 - x) equals 2/3, which again points to option A. This confirms my earlier result.Another way to think about it is using the unit circle. If sin(π/7 - x) is negative, that means the angle π/7 - x is in a quadrant where sine is negative, which would be either the third or fourth quadrant. Since π/7 is approximately 0.448 radians (about 25.7 degrees), subtracting x could place the angle in the fourth quadrant if x is positive and small, or in the third quadrant if x is larger.But regardless of the quadrant, when we take the cosine of 9π/14 - x, which is π/2 + π/7 - x, we're essentially looking at the cosine of an angle that's π/2 more than π/7 - x. Adding π/2 to an angle in the unit circle shifts it to a different quadrant, but the relationship between sine and cosine still holds as we've used in the identities above.I think I've covered multiple approaches: using the cosine addition formula, using the co-function identity, and even a brief consideration of the unit circle. All methods lead me to the same conclusion that cos(9π/14 - x) is 2/3. Therefore, the correct answer should be option A.**Final Answer**boxed{A}

answer：Okay, so I have this problem here about sets A and B, and I need to find their intersection, A ∩ B. Let me try to understand what each set is first.Set A is defined as {x | x - 1 < 2}. Hmm, that notation means "the set of all x such that x minus 1 is less than 2." So, I need to solve that inequality to find the range of x values in set A. Let me do that step by step.Starting with x - 1 < 2, I can add 1 to both sides to isolate x. That would give me x < 3. So, set A is all real numbers less than 3. In interval notation, that's (-∞, 3). Got that down.Now, set B is defined as {y | y = 2^x, x ∈ A}. Okay, so set B consists of all y values where y is equal to 2 raised to the power of x, and x is an element of set A. Since set A is all x less than 3, I need to find the range of y when x is less than 3.Let me think about the function y = 2^x. I know that 2^x is an exponential function that increases as x increases. When x is 0, y is 1. As x approaches negative infinity, y approaches 0. And as x approaches positive infinity, y approaches infinity. But in this case, x is limited to being less than 3.So, if x is less than 3, what does that mean for y? Let's see. When x is 3, y would be 2^3, which is 8. But since x is less than 3, y will be less than 8. Also, since x can be any real number less than 3, including negative numbers, y can be any positive number less than 8. So, set B is all y such that 0 < y < 8. In interval notation, that's (0, 8).Now, I need to find the intersection of sets A and B, which is A ∩ B. The intersection of two sets consists of all elements that are common to both sets. Set A is (-∞, 3) and set B is (0, 8). So, I need to find the overlap between these two intervals.Let me visualize this on a number line. Set A goes from negative infinity up to 3, and set B goes from 0 up to 8. The overlapping part where both sets are true is from 0 up to 3. Because below 0, set B doesn't include those numbers, and above 3, set A doesn't include those numbers. So, the intersection is (0, 3).Let me double-check to make sure I didn't make a mistake. Set A is x < 3, which is (-∞, 3). Set B is y = 2^x where x < 3, so y < 8 and y > 0 because 2^x is always positive. So, set B is (0, 8). The intersection would be where both conditions are satisfied, which is where x is between 0 and 3. That makes sense because for x between 0 and 3, both A and B include those values.Wait, hold on. Is the intersection in terms of x or y? The problem says A ∩ B, but set A is defined in terms of x and set B is defined in terms of y. So, are we supposed to find the intersection of the sets as numbers, regardless of whether they are x or y? Or is there a different interpretation?Hmm, that's a good point. Maybe I need to clarify. Set A is a set of x-values, and set B is a set of y-values. So, if we're talking about A ∩ B, we're looking for numbers that are in both sets, regardless of whether they were originally x or y. But x and y are different variables, so does that mean the intersection is empty? That doesn't seem right.Wait, maybe I misinterpreted the problem. Let me read it again. "Given set A = {x | x - 1 < 2} and set B = {y | y = 2^x, x ∈ A}, find A ∩ B = ( )." So, set A is a set of real numbers x, and set B is a set of real numbers y. So, A and B are both subsets of the real numbers, just defined differently.Therefore, A ∩ B would be the set of real numbers that are both in A and in B. So, numbers that are less than 3 (from A) and also numbers that are between 0 and 8 (from B). So, the overlap is numbers between 0 and 3. So, the intersection is (0, 3). That makes sense.I think my initial answer was correct. The intersection is (0, 3), which corresponds to option D. So, I feel confident with that answer.**Final Answer**boxed{D}