answer：First, I need to determine which side of the triangle is the longest. The given side lengths are 10, 24, and 26. Clearly, 26 is the longest side.Next, I should check if the triangle is a right-angled triangle by verifying the Pythagorean theorem. I'll square the lengths of the two shorter sides and see if their sum equals the square of the longest side.Calculating (10^2 + 24^2) gives (100 + 576 = 676), which is equal to (26^2). This confirms that the triangle is right-angled with the hypotenuse being 26.In a right-angled triangle, the median to the hypotenuse is equal to half the length of the hypotenuse. Therefore, the length of the median to the longest side is (26 div 2 = 13).

answer：Alright, so I have this problem about a function ( f(x) ) with some specific properties, and I need to find ( f(2014) ). Let me try to unpack each piece of information step by step.First, the problem states that for any real number ( x ), the function satisfies ( f(x + 6) + f(x) = 0 ). Hmm, that looks like a functional equation. Let me rewrite that to understand it better:[ f(x + 6) = -f(x) ]Okay, so if I shift the input by 6 units, the function's value becomes the negative of the original value. Interesting. I wonder if this implies some periodicity? Let me check by applying the equation again, but this time shifting ( x ) by another 6 units:[ f(x + 12) = -f(x + 6) ]But wait, from the original equation, I know that ( f(x + 6) = -f(x) ), so substituting that in:[ f(x + 12) = -(-f(x)) = f(x) ]Ah! So ( f(x + 12) = f(x) ), which means the function is periodic with period 12. That's a crucial piece of information. So every 12 units, the function repeats its values. That will definitely help in calculating ( f(2014) ) because I can reduce 2014 modulo 12 to find an equivalent value within one period.Next, the problem mentions that the graph of ( y = f(x - 1) ) is symmetric about the point ( (1, 0) ). I need to interpret what this means. Symmetry about a point usually means that if you rotate the graph 180 degrees around that point, it looks the same. In mathematical terms, for a function ( g(x) ), if it's symmetric about the point ( (a, b) ), then:[ g(a + h) + g(a - h) = 2b ]In this case, the function is ( y = f(x - 1) ) and it's symmetric about ( (1, 0) ). Let me denote ( g(x) = f(x - 1) ). Then, the symmetry condition translates to:[ g(1 + h) + g(1 - h) = 0 ]Substituting ( g(x) = f(x - 1) ):[ f((1 + h) - 1) + f((1 - h) - 1) = 0 ][ f(h) + f(-h) = 0 ]So, this simplifies to:[ f(h) = -f(-h) ]Which means that ( f(x) ) is an odd function. That's another important property. Odd functions satisfy ( f(-x) = -f(x) ) for all ( x ). So, knowing that ( f ) is odd will help in evaluating ( f ) at negative arguments if needed.We are also given that ( f(2) = 4 ). Since ( f ) is odd, that tells us:[ f(-2) = -f(2) = -4 ]Alright, so far I have:1. ( f ) is periodic with period 12: ( f(x + 12) = f(x) )2. ( f ) is odd: ( f(-x) = -f(x) )3. ( f(2) = 4 ) and hence ( f(-2) = -4 )Now, the question is to find ( f(2014) ). Let me see how I can use the periodicity here. Since the function repeats every 12 units, I can find an equivalent value of 2014 modulo 12. That is, I can write 2014 as:[ 2014 = 12 times k + r ]Where ( k ) is some integer and ( r ) is the remainder when 2014 is divided by 12. Then, ( f(2014) = f(r) ).Let me compute ( 2014 div 12 ):12 times 167 is 2004 (since 12*160=1920 and 12*7=84, so 1920+84=2004). Then, 2014 - 2004 = 10. So:[ 2014 = 12 times 167 + 10 ]Therefore, ( f(2014) = f(10) ).Now, I need to find ( f(10) ). Let's see if I can relate this to a known value using the properties of the function.From the functional equation ( f(x + 6) = -f(x) ), let me set ( x = 4 ):[ f(4 + 6) = -f(4) ][ f(10) = -f(4) ]So, ( f(10) = -f(4) ). Now, I need to find ( f(4) ). Let's use the functional equation again, this time with ( x = -2 ):[ f(-2 + 6) = -f(-2) ][ f(4) = -f(-2) ]But since ( f ) is odd, ( f(-2) = -f(2) = -4 ). So:[ f(4) = -(-4) = 4 ]Therefore, ( f(4) = 4 ). Plugging this back into the expression for ( f(10) ):[ f(10) = -f(4) = -4 ]So, ( f(10) = -4 ), which means ( f(2014) = -4 ).Wait a second, let me double-check my steps to make sure I didn't make a mistake.1. I found that ( f ) is periodic with period 12, so ( f(2014) = f(10) ).2. Then, using the functional equation ( f(x + 6) = -f(x) ), I set ( x = 4 ) to get ( f(10) = -f(4) ).3. To find ( f(4) ), I set ( x = -2 ) in the functional equation, getting ( f(4) = -f(-2) ).4. Since ( f ) is odd, ( f(-2) = -f(2) = -4 ), so ( f(4) = 4 ).5. Therefore, ( f(10) = -4 ).Everything seems to check out. So, ( f(2014) = -4 ).Just to be thorough, let me see if there's another way to approach this, maybe by using the periodicity and the oddness together.Since ( f ) is periodic with period 12, ( f(2014) = f(2014 mod 12) = f(10) ). As before, ( f(10) = -f(4) ). Now, if I consider ( f(4) ), since ( f ) is odd, ( f(4) = -f(-4) ). But I don't know ( f(-4) ) directly. However, using the functional equation again, let me set ( x = -4 ):[ f(-4 + 6) = -f(-4) ][ f(2) = -f(-4) ][ 4 = -f(-4) ][ f(-4) = -4 ]Therefore, ( f(4) = -f(-4) = -(-4) = 4 ), which is consistent with what I found earlier. So, ( f(10) = -4 ).Another way: Let me try to express ( f(10) ) in terms of ( f(2) ) using the functional equation. Starting from ( x = 2 ):[ f(2 + 6) = -f(2) ][ f(8) = -4 ]Then, using ( x = 8 ):[ f(8 + 6) = -f(8) ][ f(14) = -(-4) = 4 ]But ( f(14) = f(14 - 12) = f(2) = 4 ), which is consistent.Wait, but I need ( f(10) ). Let me try another approach. Starting from ( x = 4 ):[ f(4 + 6) = -f(4) ][ f(10) = -f(4) ]But I already found ( f(4) = 4 ), so ( f(10) = -4 ). Yep, same result.Alternatively, starting from ( x = -2 ):[ f(-2 + 6) = -f(-2) ][ f(4) = -(-4) = 4 ]So, again, consistent.I think I've verified this in multiple ways, so I'm confident that ( f(2014) = -4 ).**Final Answer**boxed{B}

answer：First, I recognize that the product of the number of students in each major (p, q, r, s) is given as 1365. Since there are 3 accounting majors, I substitute p with 3 in the equation.This simplifies the equation to 3 multiplied by q, r, and s equals 1365. To isolate the product of q, r, and s, I divide both sides of the equation by 3.After performing the division, I find that the product of q, r, and s is 455. Without additional information about the specific values of q, r, and s, I can only conclude that their product is 455.Therefore, the relationship between the number of students in each major is that the product of the number of finance, marketing, and strategy majors is 455.

answer：Okay, so I need to find the x-intercept of the line given by the equation 4x + 7y = 28. Hmm, I remember that the x-intercept is the point where the line crosses the x-axis. At that point, the value of y should be zero because it's on the x-axis. So, I think I should set y to 0 in the equation and solve for x.Let me write that down. If y is 0, then the equation becomes 4x + 7(0) = 28. Simplifying that, 7 times 0 is 0, so the equation is just 4x = 28. Now, to find x, I need to divide both sides of the equation by 4. So, x = 28 divided by 4, which is 7. Wait, does that make sense? If x is 7 and y is 0, then the point should be (7, 0). Let me double-check my steps. I started with the equation 4x + 7y = 28, set y to 0, which gives 4x = 28, and solving for x gives x = 7. Yeah, that seems right. I guess another way to think about it is plugging the point back into the original equation to see if it works. If I plug in x = 7 and y = 0, then 4*7 + 7*0 equals 28 + 0, which is 28. That matches the right side of the equation, so it checks out. I don't think I made any mistakes here. It was pretty straightforward once I remembered that at the x-intercept, y is zero. So, the x-intercept point is (7, 0). I feel confident about this answer now.

answer：First, I need to determine the average speed of Vehicle X. I'll denote this speed as ( V_x ) (in miles per hour).Vehicle Y starts 22 miles behind Vehicle X and needs to not only catch up but also get 23 miles ahead. This means Vehicle Y must cover a total additional distance of ( 22 + 23 = 45 ) miles compared to Vehicle X.Given that Vehicle Y travels at 45 miles per hour for 5 hours, the total distance Vehicle Y covers is:[45 text{ mph} times 5 text{ hours} = 225 text{ miles}]Since Vehicle Y covers 45 miles more than Vehicle X in the same time, Vehicle X covers:[225 text{ miles} - 45 text{ miles} = 180 text{ miles}]To find the average speed of Vehicle X, I'll divide the distance Vehicle X travels by the time:[V_x = frac{180 text{ miles}}{5 text{ hours}} = 36 text{ mph}]

answer：Okay, so I have this problem about systematic sampling, and I need to figure out the probability that each individual in a population of 1003 is selected when drawing a sample of size 50. Hmm, systematic sampling is a method where you select individuals from a population at regular intervals. I remember that it's different from simple random sampling because instead of randomly picking each individual, you pick every nth person.First, I need to recall how systematic sampling works. I think it involves determining a starting point randomly and then selecting every kth individual from the population. The value of k is usually calculated by dividing the population size by the sample size. So in this case, the population size is 1003, and the sample size is 50. Let me write that down:Population size (N) = 1003 Sample size (n) = 50So, the interval k would be N divided by n, which is 1003 divided by 50. Let me calculate that:k = 1003 / 50 = 20.06Wait, that's not a whole number. I think in systematic sampling, k should be an integer. Maybe I need to round it or adjust somehow. Perhaps I can use 20 as the interval and see how that works. If I take every 20th individual, starting from a random point between 1 and 20, that should give me a sample size close to 50. But 20 times 50 is 1000, which is just 3 less than the population size of 1003. So, if I start at a random point between 1 and 20, and then take every 20th individual, I might end up with 50 individuals, but I have to make sure that the last few individuals are also included.Alternatively, maybe I can use a fractional interval. I've heard of methods where you can use a decimal interval and then randomly select the starting point within that interval. So, if k is 20.06, I could treat it as 20 and 1/16 (since 0.06 is approximately 1/16). That might complicate things, but perhaps it's necessary to ensure that the sample size is exactly 50.But wait, the question is about the probability of each individual being selected, not necessarily about the exact mechanics of selecting the sample. Maybe I don't need to worry too much about the interval calculation right now. Instead, I should focus on the probability aspect.In systematic sampling, each individual has an equal chance of being selected, right? So, if the population size is 1003 and the sample size is 50, the probability should be the ratio of the sample size to the population size. That would be 50 divided by 1003. Let me check that:Probability = n / N = 50 / 1003That seems straightforward. But I want to make sure I'm not missing something. In systematic sampling, sometimes the starting point is chosen randomly, and then every kth individual is selected. If the population size isn't a multiple of the sample size, there might be some adjustment needed to ensure that the sample size is exactly 50. But regardless of the adjustments, the probability should still be the same for each individual because the method is designed to be fair and unbiased.Let me think about it another way. If I have 1003 individuals and I need to select 50, the chance that any one individual is selected should be 50 out of 1003. That makes sense because each individual has an equal opportunity to be in the sample. It's similar to simple random sampling in that regard, where each individual has the same probability of being selected.I also recall that in systematic sampling, if the population has a periodic pattern that aligns with the sampling interval, it can introduce bias. But the question doesn't mention anything about the population having such patterns, so I think we can assume that the sampling is done fairly without any periodicity issues.So, putting it all together, the probability that each individual is selected is 50 divided by 1003. Looking back at the options provided:A: 1/1000 B: 1/1003 C: 50/1000 D: 50/1003The correct answer should be D: 50/1003. This matches my calculation and reasoning about the probability in systematic sampling.Just to double-check, if I consider the total number of possible samples, which is the number of ways to choose 50 individuals out of 1003, each individual has 50 chances out of 1003 to be selected. That aligns with the probability I calculated earlier. So, I'm confident that D is the right choice.