💡Okay, so I have this problem with vectors OA, OB, and OC given. I need to solve two parts: first, find the condition on m so that points A, B, and C form a triangle, and second, find m such that triangle ABC is right-angled at A.Starting with part (1). Hmm, for three points to form a triangle, they must not be collinear. That means the vectors AB and AC shouldn't be scalar multiples of each other. So, let me find vectors AB and AC.Vector OA is (3, -4), OB is (6, -3), and OC is (5 - m, -(3 + m)). So, vector AB is OB - OA, which is (6 - 3, -3 - (-4)) = (3, 1). Similarly, vector AC is OC - OA, which is (5 - m - 3, -(3 + m) - (-4)) = (2 - m, 1 - m).Now, if AB and AC are collinear, then there exists a scalar k such that AB = k * AC. So, (3, 1) = k*(2 - m, 1 - m). This gives two equations:3 = k*(2 - m)1 = k*(1 - m)From the second equation, k = 1 / (1 - m). Plugging this into the first equation: 3 = [1 / (1 - m)]*(2 - m). Multiply both sides by (1 - m):3*(1 - m) = 2 - m3 - 3m = 2 - m3 - 2 = 3m - m1 = 2mm = 1/2So, if m = 1/2, points A, B, C are collinear and don't form a triangle. Therefore, the condition is m ≠ 1/2.Moving on to part (2). Triangle ABC is right-angled at A, so vectors AB and AC must be perpendicular. That means their dot product should be zero.We already have AB = (3, 1) and AC = (2 - m, 1 - m). So, their dot product is:3*(2 - m) + 1*(1 - m) = 06 - 3m + 1 - m = 07 - 4m = 04m = 7m = 7/4Wait, let me double-check that. 3*(2 - m) is 6 - 3m, and 1*(1 - m) is 1 - m. Adding them: 6 + 1 - 3m - m = 7 - 4m. Setting equal to zero: 7 - 4m = 0, so 4m = 7, m = 7/4. Yep, that seems right.So, for part (1), m ≠ 1/2, and for part (2), m = 7/4.**Final Answer**(1) The condition is boxed{m neq dfrac{1}{2}}.(2) The value of ( m ) is boxed{dfrac{7}{4}}.

💡Okay, so I need to find the area of a triangle with vertices at (-2,3), (6,1), and (10,6). Hmm, I remember there are a few ways to calculate the area of a triangle when you have the coordinates of the vertices. One method is using the shoelace formula, which I think involves multiplying coordinates in a specific way and then taking half the absolute difference. Another method might be using vectors and determinants, which I think is similar to what was shown in the previous example. Let me try to recall both methods and see which one I can apply here.First, let me write down the coordinates clearly:Point A: (-2, 3)Point B: (6, 1)Point C: (10, 6)I think the shoelace formula is a straightforward method for this. The formula is:Area = (1/2) * |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))|Let me plug in the coordinates into this formula. So, assigning:x1 = -2, y1 = 3x2 = 6, y2 = 1x3 = 10, y3 = 6Plugging into the formula:Area = (1/2) * | (-2)(1 - 6) + 6(6 - 3) + 10(3 - 1) |Let me compute each term step by step.First term: (-2)(1 - 6) = (-2)(-5) = 10Second term: 6(6 - 3) = 6*3 = 18Third term: 10(3 - 1) = 10*2 = 20Now, add these up: 10 + 18 + 20 = 48Take the absolute value, which is still 48, then multiply by 1/2: (1/2)*48 = 24So, the area is 24. Hmm, that seems straightforward. But wait, let me double-check using another method to make sure I didn't make a mistake.Another method is using vectors and the cross product. I think this is similar to what was done in the initial example. Let me try that.First, I need to define vectors from one point to the other two points. Let's choose point C as the common vertex, so vectors CA and CB.Vector CA is from C to A: A - C = (-2 - 10, 3 - 6) = (-12, -3)Vector CB is from C to B: B - C = (6 - 10, 1 - 6) = (-4, -5)The area of the parallelogram formed by these vectors is the absolute value of the determinant of the matrix formed by these vectors. The determinant is calculated as:| (-12)(-5) - (-3)(-4) | = |60 - 12| = |48| = 48Since the area of the triangle is half the area of the parallelogram, we divide by 2: 48 / 2 = 24Okay, so both methods give me the same result, 24. That makes me more confident that the area is indeed 24.Wait, just to be thorough, let me visualize the triangle. Point A is at (-2,3), which is in the second quadrant. Point B is at (6,1), which is in the first quadrant, and Point C is at (10,6), also in the first quadrant. So, the triangle is spread across the second and first quadrants. I can imagine it's a bit stretched out, but the area seems reasonable.Alternatively, I could also use the base-height formula. To do that, I need to find the length of one side as the base and then find the height corresponding to that base.Let me pick side AB as the base. First, I need the length of AB.Point A: (-2,3)Point B: (6,1)Distance formula: sqrt[(x2 - x1)^2 + (y2 - y1)^2]So, AB = sqrt[(6 - (-2))^2 + (1 - 3)^2] = sqrt[(8)^2 + (-2)^2] = sqrt[64 + 4] = sqrt[68] ≈ 8.246Now, I need the height corresponding to this base. The height is the perpendicular distance from point C to the line AB.First, let me find the equation of line AB.Slope of AB: (y2 - y1)/(x2 - x1) = (1 - 3)/(6 - (-2)) = (-2)/8 = -1/4So, the slope of AB is -1/4. Therefore, the equation of AB can be written as:y - y1 = m(x - x1)Using point A (-2,3):y - 3 = (-1/4)(x - (-2)) => y - 3 = (-1/4)(x + 2)Simplify:y = (-1/4)x - (1/4)*2 + 3y = (-1/4)x - 0.5 + 3y = (-1/4)x + 2.5So, the equation of line AB is y = (-1/4)x + 2.5Now, the distance from point C (10,6) to this line is the height.The formula for the distance from a point (x0, y0) to the line ax + by + c = 0 is:Distance = |a*x0 + b*y0 + c| / sqrt(a^2 + b^2)First, let me write the equation of AB in standard form:y = (-1/4)x + 2.5Multiply both sides by 4 to eliminate the fraction:4y = -x + 10Bring all terms to one side:x + 4y - 10 = 0So, a = 1, b = 4, c = -10Now, plug in point C (10,6):Distance = |1*10 + 4*6 -10| / sqrt(1^2 + 4^2) = |10 + 24 -10| / sqrt(1 + 16) = |24| / sqrt(17) = 24 / sqrt(17)So, the height is 24 / sqrt(17)Now, area = (1/2)*base*height = (1/2)*(sqrt(68))*(24 / sqrt(17))Simplify sqrt(68): sqrt(4*17) = 2*sqrt(17)So, base = 2*sqrt(17)Thus, area = (1/2)*(2*sqrt(17))*(24 / sqrt(17)) = (sqrt(17))*(24 / sqrt(17)) = 24So, again, the area is 24. That's the third method, and it still gives me the same result. I think I'm pretty confident now that 24 is the correct area.Just to recap, I used three different methods:1. Shoelace formula: Got 242. Vectors and determinant: Got 243. Base-height with distance formula: Got 24All methods led to the same answer, so I don't think I made a calculation error. It's always good to cross-verify using different methods, especially when dealing with coordinate geometry.I guess another way to think about it is plotting the points on a graph. If I were to plot (-2,3), (6,1), and (10,6), I could approximate the area, but since all the calculations give me 24, I think that's solid.Wait, just to make sure, maybe I can try using vectors from a different point. Let's say, instead of taking point C as the common vertex, take point A.So, vectors AB and AC.Vector AB = B - A = (6 - (-2), 1 - 3) = (8, -2)Vector AC = C - A = (10 - (-2), 6 - 3) = (12, 3)Compute the determinant:| (8)(3) - (-2)(12) | = |24 - (-24)| = |24 + 24| = |48| = 48Area of triangle is half that, so 24. Yep, same result.Alternatively, using point B as the common vertex.Vectors BA and BC.Vector BA = A - B = (-2 - 6, 3 - 1) = (-8, 2)Vector BC = C - B = (10 - 6, 6 - 1) = (4, 5)Determinant:| (-8)(5) - (2)(4) | = |-40 - 8| = |-48| = 48Area is 48 / 2 = 24. Same answer.So, regardless of which point I take as the common vertex, the area remains the same. That makes sense because the area of the triangle shouldn't depend on the choice of vectors, just on the relative positions of the points.I think I've covered all the bases here. I've used multiple methods, double-checked each calculation, and all lead to the same conclusion. I don't see any mistakes in my reasoning, so I feel confident that the area is indeed 24.**Final Answer**The area of the triangle is boxed{24}.

💡First, I need to determine the discounted price per can when purchasing in 24-can cases. The regular price per can is 0.15, and there's a 10% discount for buying in cases.To find the discount amount, I'll calculate 10% of 0.15:0.10 * 0.15 = 0.015Subtracting this discount from the regular price gives the discounted price per can:0.15 - 0.015 = 0.135Next, I need to calculate the total price for 75 cans. Since the discount applies only to full 24-can cases, I'll determine how many full cases can be purchased and how many individual cans remain.Dividing 75 by 24:75 ÷ 24 = 3 with a remainder of 3This means I can buy 3 full cases (72 cans) at the discounted price and 3 individual cans at the regular price.Calculating the cost for the 3 full cases:72 cans * 0.135 per can = 9.72Calculating the cost for the remaining 3 cans:3 cans * 0.15 per can = 0.45Adding both amounts gives the total price:9.72 + 0.45 = 10.17Therefore, the total price for 75 cans purchased in 24-can cases is 10.17.

💡Okay, so I have this problem where several classmates together ate a cake. Alex ate the most, which is 1/11 of the entire cake, and Alena ate the least, which is 1/14 of the entire cake. I need to figure out how many classmates ate the cake and list all possible options.Hmm, let me start by understanding the problem. There are multiple people sharing a cake. Alex ate the largest portion, which is 1/11, and Alena ate the smallest portion, which is 1/14. So, the total number of people must be somewhere between these two fractions. First, let me think about the minimum number of people. If Alex ate 1/11, that suggests there could be 11 people if everyone ate the same amount. But since Alex ate the most, there must be more than 11 people because if there were only 11, everyone would have to eat at least 1/11, but Alex ate the most, so others must have eaten less. Wait, no, actually, if Alex ate 1/11, the others could have eaten less, so maybe it's not necessarily more than 11. Hmm, I need to think carefully.Similarly, Alena ate the least, which is 1/14. So, if Alena ate 1/14, the total number of people can't be more than 14 because if there were 14 people, each would have to eat at least 1/14, but Alena ate the least, so others could have eaten more. So, the number of people must be between 12 and 13? Wait, let me check.Let me denote the number of classmates as n. So, n must satisfy that 1/11 is the largest portion, and 1/14 is the smallest portion. So, each person's portion must be between 1/14 and 1/11.Therefore, the total cake eaten by all classmates is 1. So, the sum of all portions must be 1.Let me write that as an equation. Let’s say each person’s portion is a fraction, and the sum of all these fractions is 1.So, if we have n classmates, each eating between 1/14 and 1/11, then the total sum is 1.But since Alex ate the most, which is 1/11, and Alena ate the least, which is 1/14, the remaining n-2 classmates must have eaten portions between 1/14 and 1/11.So, let me denote the portions as follows:- Alex: 1/11- Alena: 1/14- Others: x_i, where each x_i is between 1/14 and 1/11, for i = 3 to n.So, the total sum is:1/11 + 1/14 + sum_{i=3}^n x_i = 1Therefore, sum_{i=3}^n x_i = 1 - 1/11 - 1/14Let me compute that:1 - 1/11 - 1/14First, find a common denominator for 11 and 14, which is 154.So, 1 = 154/1541/11 = 14/1541/14 = 11/154So,1 - 1/11 - 1/14 = 154/154 - 14/154 - 11/154 = (154 - 14 - 11)/154 = (154 - 25)/154 = 129/154So, the sum of the remaining portions is 129/154.Now, the number of remaining classmates is n - 2.Each of these n - 2 classmates ate between 1/14 and 1/11.So, the total sum of their portions is 129/154.Therefore, we have:(n - 2) * (1/14) ≤ 129/154 ≤ (n - 2) * (1/11)Because each x_i is at least 1/14 and at most 1/11, so the total sum is at least (n - 2)/14 and at most (n - 2)/11.So, let's write these inequalities:(n - 2)/14 ≤ 129/154 ≤ (n - 2)/11Let me solve these inequalities for n.First, let's solve the left inequality:(n - 2)/14 ≤ 129/154Multiply both sides by 14:n - 2 ≤ (129/154) * 14Simplify the right side:129/154 * 14 = 129/11 ≈ 11.727So,n - 2 ≤ 11.727Therefore,n ≤ 13.727Since n must be an integer, n ≤ 13.Now, let's solve the right inequality:129/154 ≤ (n - 2)/11Multiply both sides by 11:129/154 * 11 ≤ n - 2Simplify the left side:129/154 * 11 = 129/14 ≈ 9.214So,9.214 ≤ n - 2Therefore,n ≥ 11.214Since n must be an integer, n ≥ 12.So, combining both inequalities, n must be between 12 and 13 inclusive.Therefore, the possible number of classmates is 12 or 13.Wait, let me verify if both 12 and 13 are possible.For n = 12:Total remaining portions: 129/154Number of remaining classmates: 12 - 2 = 10So, each of the remaining 10 classmates ate (129/154)/10 = 129/1540 ≈ 0.0837Now, check if this is between 1/14 ≈ 0.0714 and 1/11 ≈ 0.0909.Yes, 0.0714 < 0.0837 < 0.0909, so it's valid.For n = 13:Total remaining portions: 129/154Number of remaining classmates: 13 - 2 = 11So, each of the remaining 11 classmates ate (129/154)/11 = 129/1694 ≈ 0.0761Check if this is between 1/14 ≈ 0.0714 and 1/11 ≈ 0.0909.Yes, 0.0714 < 0.0761 < 0.0909, so it's also valid.Therefore, both 12 and 13 are possible numbers of classmates.So, the possible options are 12 or 13.

💡Okay, so I need to find a quadratic surd that's similar to √2. Hmm, first, what exactly is a quadratic surd? I think it's an expression that involves the square root of a non-square integer, right? So, √2 is a quadratic surd because 2 isn't a perfect square. Now, what does it mean for two quadratic surds to be similar? I guess they should have the same radicand, which is the number under the square root. So, if I have √2, a similar surd would also involve √2 but maybe multiplied by some rational number. Wait, but the problem just says "similar," not "equivalent" or "equal." So, maybe it's about having the same form or structure. Maybe it's about having the same type of expression, like a multiple of √2. Let me think. If I take √2 and multiply it by any rational number, say 3, I get 3√2. Is that considered similar? I think so because it's still a multiple of √2, just scaled by a rational factor. But the problem didn't specify any particular multiple, so I guess any multiple would work. Maybe I should choose a simple one to make it easy. Let's pick 2. So, 2√2 would be similar to √2. Wait, is there a specific definition of "similar" in this context? Maybe it's about having the same minimal polynomial or something like that. The minimal polynomial of √2 is x² - 2. If I take 2√2, its minimal polynomial would be x² - 8, right? Because (2√2)² = 8. So, they have different minimal polynomials. Does that matter? Hmm, maybe "similar" just means having the same radicand, regardless of the coefficient. So, as long as it's a multiple of √2, it's similar. So, 2√2, 3√2, etc., are all similar to √2. Alternatively, maybe "similar" refers to being in the same equivalence class under some relation. But I'm not sure about that. I think for this problem, it's probably simpler. They just want another quadratic surd that involves √2 multiplied by a rational number. So, to keep it simple, I'll go with 2√2 as the similar quadratic surd to √2. It's straightforward and meets the basic requirement of being a multiple of √2. But just to double-check, let's see if there's another way to interpret "similar." Maybe it's about having the same properties, like being irrational and not expressible as a fraction. Well, any multiple of √2 by a non-zero rational number would still be irrational, so that fits. Okay, I think I'm confident enough that 2√2 is a suitable answer. It's similar to √2 because it's just √2 scaled by a rational factor, and it maintains the properties of a quadratic surd.

💡First, I need to calculate the time taken for each section of the journey. For the flat surface, the cyclist travels 16 meters at an average speed of 8 meters per second. Next, the cyclist ascends a 12-meter slope at a speed of 6 meters per second. Finally, the cyclist descends a 20-meter slope at a speed of 12 meters per second. After finding the time for each section, I will add them up to get the total time taken for the entire journey. With the total distance and total time, I can then calculate the average speed by dividing the total distance by the total time.