💡Alright, so I have this problem where I'm given three points in space: A(0,0,1), B(-1,1,1), and C(1,2,-3). I need to find a point M on the line AB such that the vector CM is perpendicular to AB. Hmm, okay, let's break this down step by step.First, I should probably visualize the points. Point A is at (0,0,1), which is on the z-axis. Point B is at (-1,1,1), so it's a bit to the left in the x-direction and up in the y-direction, but still at the same z-coordinate as A. Point C is at (1,2,-3), which is further out in the x and y directions but much lower in the z-direction.Now, I need to find point M on line AB such that CM is perpendicular to AB. Okay, so M lies somewhere between A and B (or maybe even beyond, depending on the parametrization). Since M is on AB, I can express its coordinates in terms of a parameter.Let me recall that a point on a line can be expressed parametrically. So, the line AB can be parametrized as A + t*(B - A), where t is a scalar parameter. Let me compute vector AB first.Vector AB is B - A, which is (-1 - 0, 1 - 0, 1 - 1) = (-1, 1, 0). So, the parametric equations for line AB would be:x = 0 + (-1)*t = -ty = 0 + 1*t = tz = 1 + 0*t = 1So, any point M on AB can be written as M(-t, t, 1), where t is a real number. Okay, that seems straightforward.Now, I need to find the specific value of t such that vector CM is perpendicular to vector AB. Let's recall that two vectors are perpendicular if their dot product is zero.First, let's find vector CM. Vector CM is M - C. Since M is (-t, t, 1) and C is (1, 2, -3), vector CM is:(-t - 1, t - 2, 1 - (-3)) = (-t - 1, t - 2, 4)Okay, so vector CM is (-t - 1, t - 2, 4).Vector AB, as we found earlier, is (-1, 1, 0).Now, the dot product of CM and AB should be zero:(-t - 1)*(-1) + (t - 2)*(1) + (4)*(0) = 0Let me compute each term:First term: (-t - 1)*(-1) = t + 1Second term: (t - 2)*(1) = t - 2Third term: 4*0 = 0So, adding them up: (t + 1) + (t - 2) + 0 = 0Simplify:t + 1 + t - 2 = 0Combine like terms:2t - 1 = 0So, 2t = 1 => t = 1/2Okay, so t is 1/2. Now, let's plug this back into the parametric equations for M.x = -t = -1/2y = t = 1/2z = 1So, point M is (-1/2, 1/2, 1)Wait, let me double-check my calculations to make sure I didn't make a mistake.First, vector AB is (-1, 1, 0), correct.Parametric equations for M: (-t, t, 1), that seems right.Vector CM: M - C = (-t - 1, t - 2, 4), yes.Dot product with AB: (-t - 1)*(-1) + (t - 2)*(1) + 4*0.Compute:(-t -1)*(-1) = t + 1(t - 2)*1 = t - 2Sum: t + 1 + t - 2 = 2t -1Set to zero: 2t -1 =0 => t=1/2Plugging back in: x=-1/2, y=1/2, z=1. Seems correct.So, the coordinates of point M are (-1/2, 1/2, 1). I think that's the answer.Just to be thorough, let me verify that vector CM is indeed perpendicular to AB.Vector CM is M - C = (-1/2 -1, 1/2 -2, 1 - (-3)) = (-3/2, -3/2, 4)Vector AB is (-1,1,0)Dot product: (-3/2)*(-1) + (-3/2)*(1) + 4*0 = 3/2 - 3/2 + 0 = 0Yes, the dot product is zero, so they are perpendicular. That checks out.Therefore, I'm confident that the coordinates of point M are (-1/2, 1/2, 1).**Final Answer**The coordinates of point ( M ) are boxed{left( -dfrac{1}{2}, dfrac{1}{2}, 1 right)}.

💡Alright, so I have this problem about a large rectangle that's been divided into smaller rectangles. The total area of the large rectangle is 72 square units. There are five regions in total, and three of their areas are given: 15, 12, and 18 square units. I need to find the area of another undivided smaller rectangle that's not listed among these given areas.First, I need to visualize how the large rectangle is divided. It says it's divided by two lines, one parallel to its length and the other to its width. So, if I imagine the large rectangle, there's a line drawn parallel to its length, which would divide it into two smaller rectangles vertically, and another line parallel to its width, which would divide it into two smaller rectangles horizontally. This creates four smaller rectangles.Then, one of these smaller rectangles is itself divided by a line parallel to the length of the original rectangle. So, this means that one of the four smaller rectangles is further divided into two even smaller rectangles, making a total of five regions.Let me try to sketch this mentally. Suppose the large rectangle is divided vertically into two parts, say left and right, and horizontally into two parts, say top and bottom. This creates four smaller rectangles: top-left, top-right, bottom-left, and bottom-right. Now, if one of these four is further divided vertically, it will create two more regions. So, depending on which one is divided, the areas will change accordingly.Given that the total area is 72, and three of the five regions have areas 15, 12, and 18, I need to figure out the area of the fourth region.Let me denote the large rectangle as ABCD, with AB and CD as the lengths, and AD and BC as the widths. Let's say we draw a vertical line from the top, dividing the rectangle into two parts: left and right. Similarly, we draw a horizontal line from the left, dividing the rectangle into top and bottom. This gives us four smaller rectangles: top-left (let's say area 15), top-right, bottom-left, and bottom-right.Now, one of these four is further divided vertically. Suppose the bottom-right rectangle is divided vertically, creating two smaller regions: one with area 12 and another with area 18. So, the bottom-right rectangle is split into two, making the total regions five: top-left (15), top-right, bottom-left, bottom-right-left (12), and bottom-right-right (18).Wait, but the problem states that one of the smaller rectangles is divided by a line parallel to the length of the original rectangle. So, if the original rectangle has length AB, then the line parallel to AB would be horizontal. But in my previous thought, I considered a vertical division. Maybe I need to clarify this.If the line is parallel to the length, which is AB, then it's a horizontal line. So, if one of the smaller rectangles is divided by a horizontal line, it would create two regions stacked vertically within that smaller rectangle.Let me adjust my visualization. Suppose the large rectangle is divided into four smaller rectangles by a vertical line and a horizontal line. Then, one of these four is further divided by another horizontal line, creating two more regions within it. So, in total, we have five regions.Given that, let's assign areas to these regions. Let's say the top-left rectangle has area 15, the top-right has area 12, the bottom-left has area 18, and the bottom-right is divided into two regions. The total area is 72, so the sum of all five regions should be 72.Wait, but the problem says that three of the regions have areas 15, 12, and 18, and we need to find the area of another undivided smaller rectangle. So, perhaps the bottom-right rectangle is undivided, and the other four regions include the three given areas and the divided part.I think I need to set up some variables to represent the dimensions and areas.Let me denote the length of the large rectangle as L and the width as W, so the area is L * W = 72.Now, when we divide the large rectangle with a vertical line, we create two smaller rectangles with widths w1 and w2, where w1 + w2 = W. Similarly, dividing it with a horizontal line creates two smaller rectangles with lengths l1 and l2, where l1 + l2 = L.So, the four smaller rectangles have areas:1. Top-left: l1 * w12. Top-right: l1 * w23. Bottom-left: l2 * w14. Bottom-right: l2 * w2Now, one of these four is further divided by a line parallel to the length, which is horizontal. So, suppose we divide the bottom-right rectangle (l2 * w2) into two smaller rectangles with lengths l3 and l4, where l3 + l4 = l2. So, the areas of these two smaller rectangles would be l3 * w2 and l4 * w2.Given that, the total areas of the five regions would be:1. Top-left: l1 * w1 = 152. Top-right: l1 * w2 = 123. Bottom-left: l2 * w1 = 184. Bottom-right-left: l3 * w25. Bottom-right-right: l4 * w2We need to find the area of another undivided smaller rectangle, which would be either the bottom-right-left or bottom-right-right, but since they are divided, perhaps the undivided one is the bottom-right before division, but that's not possible because it's already divided. Wait, maybe I misassigned the areas.Alternatively, perhaps the undivided smaller rectangle is one of the top or bottom ones, but they are already given. Hmm.Wait, the problem says that one of the smaller rectangles is itself divided by a line parallel to the length, making a total of five regions. So, originally, there were four smaller rectangles, and one of them is divided into two, making five. So, the undivided smaller rectangles are three, and the divided one is split into two, making five regions.Given that, the three given areas are among the five regions, and we need to find the area of another undivided smaller rectangle, which is one of the original four before division.So, perhaps the three given areas include the two parts of the divided rectangle and two of the undivided ones, and we need to find the third undivided one.Let me try to set up equations.Let me denote:- Top-left: A = l1 * w1 = 15- Top-right: B = l1 * w2 = 12- Bottom-left: C = l2 * w1 = 18- Bottom-right: D = l2 * w2Now, one of these four is divided into two parts. Suppose D is divided into D1 and D2, so D = D1 + D2.Given that, the total area is A + B + C + D1 + D2 = 72.But since D = D1 + D2, we have A + B + C + D = 72.Given A = 15, B = 12, C = 18, so 15 + 12 + 18 + D = 72 => D = 72 - 45 = 27.So, D = 27. But D is divided into D1 and D2, which are two regions. However, the problem states that three of the regions have areas 15, 12, and 18, and we need to find the area of another undivided smaller rectangle.Wait, but if D is divided into D1 and D2, then the five regions are A, B, C, D1, D2. Among these, three are given: A=15, B=12, C=18. So, D1 and D2 are the other two regions, but their areas are not given. However, the problem says that three of the regions have areas 15, 12, and 18, and we need to find the area of another undivided smaller rectangle.Wait, perhaps the undivided smaller rectangles are A, B, C, and D, but D is divided, so the undivided ones are A, B, and C. But A, B, and C are already given as 15, 12, and 18. So, that doesn't make sense.Alternatively, maybe the undivided smaller rectangles are A, B, C, and D, but D is divided, so the undivided ones are A, B, and C. But again, their areas are given.Wait, perhaps I misassigned which rectangle is divided. Maybe it's not D that's divided, but one of the others.Suppose instead that the top-right rectangle B is divided into two parts. So, B = B1 + B2, where B1 and B2 are the areas of the two smaller rectangles created by the division.Given that, the five regions would be A, B1, B2, C, and D.Given that, three of these are A=15, C=18, and either B1 or B2, but the problem states that three areas are 15, 12, and 18. So, if A=15, C=18, and B1=12, then B2 would be B - B1 = 12 - 12 = 0, which doesn't make sense. Alternatively, if B1=12 and B2= something else.Wait, perhaps B is divided into B1=12 and B2= something, but then the total area would be A=15, B1=12, B2=?, C=18, D=?But the total area is 72, so 15 + 12 + B2 + 18 + D = 72 => B2 + D = 27.But we don't know B2 or D. Hmm.Alternatively, maybe the bottom-left rectangle C is divided into two parts. So, C = C1 + C2.Then, the five regions are A=15, B=12, C1, C2, D.Given that, three areas are 15, 12, and 18. So, either C1=18 or C2=18, but then the other would be C - 18.But C = l2 * w1 = 18, so if C is divided into C1 and C2, then C1 + C2 = 18. So, if one of them is 18, the other would be 0, which is not possible. So, that can't be.Alternatively, maybe the top-left rectangle A is divided into two parts. So, A = A1 + A2.Then, the five regions are A1, A2, B=12, C=18, D.Given that, three areas are 15, 12, and 18. So, either A1=15 or A2=15, but then the other would be A - 15.But A = l1 * w1 = 15, so if A is divided into A1 and A2, then A1 + A2 = 15. So, if one of them is 15, the other would be 0, which is not possible. So, that can't be.Wait, maybe I need to consider that the division is not necessarily along the same line as the initial division. Maybe the line parallel to the length is drawn in a different way.Alternatively, perhaps the large rectangle is divided into four smaller rectangles by one vertical and one horizontal line, and then one of these four is divided by another line parallel to the length, which could be either vertical or horizontal, but the problem says parallel to the length of the original rectangle, which is vertical. Wait, no, the length is horizontal, so parallel to the length would be horizontal.Wait, the problem says: "one of these smaller rectangles is itself divided by a line parallel to the length of the original rectangle." So, the original rectangle has length AB, which is horizontal, so a line parallel to AB would be horizontal. Therefore, the division is horizontal.So, if one of the four smaller rectangles is divided by a horizontal line, it would create two smaller rectangles stacked vertically within that smaller rectangle.Given that, let's suppose that the bottom-right rectangle D is divided into two parts: D1 and D2, with D1 on top and D2 at the bottom.So, the five regions are A=15, B=12, C=18, D1, D2.Given that, the total area is 15 + 12 + 18 + D1 + D2 = 72 => D1 + D2 = 27.But we need to find the area of another undivided smaller rectangle, which would be D, but D is divided into D1 and D2. So, perhaps the undivided smaller rectangles are A, B, and C, but their areas are already given.Wait, no, the undivided smaller rectangles are A, B, C, and D, but D is divided, so the undivided ones are A, B, and C. But their areas are already given as 15, 12, and 18. So, that doesn't make sense.Wait, maybe I need to consider that the division is not necessarily in the bottom-right rectangle. Maybe it's in another one.Suppose the top-right rectangle B is divided into two parts by a horizontal line. So, B = B1 + B2.Then, the five regions are A=15, B1, B2, C=18, D.Given that, three areas are 15, 12, and 18. So, either B1=12 or B2=12.If B1=12, then B2 = B - 12. But B = l1 * w2 = 12, so B2 = 12 - 12 = 0, which is not possible.If B2=12, then B1 = B - 12 = 12 - 12 = 0, which is also not possible.So, that can't be.Alternatively, suppose the bottom-left rectangle C is divided into two parts by a horizontal line. So, C = C1 + C2.Then, the five regions are A=15, B=12, C1, C2, D.Given that, three areas are 15, 12, and 18. So, either C1=18 or C2=18.But C = l2 * w1 = 18, so if C is divided into C1 and C2, then C1 + C2 = 18. So, if one of them is 18, the other would be 0, which is not possible.Alternatively, maybe the top-left rectangle A is divided into two parts by a horizontal line. So, A = A1 + A2.Then, the five regions are A1, A2, B=12, C=18, D.Given that, three areas are 15, 12, and 18. So, either A1=15 or A2=15.But A = l1 * w1 = 15, so if A is divided into A1 and A2, then A1 + A2 = 15. So, if one of them is 15, the other would be 0, which is not possible.Hmm, this is confusing. Maybe I need to approach this differently.Let me consider the ratios of the areas.From the top-left and top-right rectangles, A=15 and B=12, which share the same length l1. So, the ratio of their widths w1 and w2 is 15:12, which simplifies to 5:4.Similarly, from the top-left and bottom-left rectangles, A=15 and C=18, which share the same width w1. So, the ratio of their lengths l1 and l2 is 15:18, which simplifies to 5:6.So, we have:- w1:w2 = 5:4- l1:l2 = 5:6Let me denote w1 = 5k, w2 = 4k, l1 = 5m, l2 = 6m.Then, the areas are:- A = l1 * w1 = 5m * 5k = 25mk = 15- B = l1 * w2 = 5m * 4k = 20mk = 12- C = l2 * w1 = 6m * 5k = 30mk = 18- D = l2 * w2 = 6m * 4k = 24mkFrom A = 25mk = 15 => mk = 15/25 = 3/5.From B = 20mk = 12 => mk = 12/20 = 3/5. Consistent.From C = 30mk = 18 => mk = 18/30 = 3/5. Consistent.So, mk = 3/5.Therefore, D = 24mk = 24*(3/5) = 72/5 = 14.4.Wait, but D is supposed to be divided into two parts, D1 and D2, making the total area of D1 + D2 = 14.4.But the problem states that three of the regions have areas 15, 12, and 18, and we need to find the area of another undivided smaller rectangle.Wait, but according to this, D is 14.4, which is not among the given areas. So, perhaps the undivided smaller rectangle is D, which is 14.4, but that's not one of the options.Wait, the options are 24, 27, 30, 36, 40. So, 14.4 is not among them. Hmm.Maybe I made a mistake in assigning which rectangle is divided.Alternatively, perhaps the division is not in D, but in another rectangle.Wait, let's see. If D is 14.4, and it's divided into two parts, D1 and D2, then the total area of D1 + D2 = 14.4.But the problem states that three of the regions have areas 15, 12, and 18, which are already accounted for by A, B, and C. So, the fifth region would be either D1 or D2, but their areas are not given.Wait, but the problem says that the areas of three of the regions are 15, 12, and 18, and the total area is 72. So, the sum of the three given areas is 15 + 12 + 18 = 45. Therefore, the sum of the remaining two regions is 72 - 45 = 27.So, the two divided regions sum up to 27. But the problem asks for the area of another undivided smaller rectangle, which would be D, but D is divided into two parts, so D is not undivided.Wait, perhaps I need to consider that the undivided smaller rectangles are A, B, C, and D, but D is divided, so the undivided ones are A, B, and C, which are already given. So, that can't be.Alternatively, maybe the undivided smaller rectangle is one of the divided parts, but that doesn't make sense.Wait, perhaps I need to consider that the division is not in D, but in another rectangle, and thus the undivided smaller rectangle is D.Wait, let's try this again.If the division is in D, then D is divided into D1 and D2, and the total area of D is 14.4, as calculated earlier. But the problem states that three of the regions have areas 15, 12, and 18, which sum to 45, leaving 27 for the other two regions, which would be D1 and D2.But according to my earlier calculation, D1 + D2 = 14.4, which contradicts the total area.Wait, that must mean that my initial assumption about which rectangle is divided is incorrect.Perhaps the division is not in D, but in another rectangle, say B.If B is divided into B1 and B2, then the total area of B is 12, so B1 + B2 = 12.Given that, the total area of the five regions would be A=15, B1, B2, C=18, D=?Sum: 15 + B1 + B2 + 18 + D = 72 => 33 + 12 + D = 72 => D = 27.So, D = 27, which is one of the options.Therefore, the area of the undivided smaller rectangle D is 27.Wait, but D is supposed to be divided, right? Because the problem says that one of the smaller rectangles is divided, making five regions.Wait, no, if B is divided, then D remains undivided. So, the undivided smaller rectangles are A, C, and D, with areas 15, 18, and 27, respectively.But the problem states that three of the regions have areas 15, 12, and 18, which are A=15, B1 or B2=12, and C=18. Then, the undivided smaller rectangle D has area 27.Yes, that makes sense.So, the area of the undivided smaller rectangle D is 27, which is option B.Therefore, the answer is 27 square units.

💡Okay, so I have this problem where I need to compute M times the vector combination 3u minus v plus 2w. I know that M is a matrix, and u, v, w are vectors. They've given me the results of M multiplied by each of these vectors individually. Let me write down what I know:M*u = [3, -1]^TM*v = [-2, 4]^TM*w = [5, -3]^TAnd I need to find M*(3u - v + 2w). Hmm, okay. I remember that matrix multiplication is linear, which means I can distribute M over the addition and subtraction. So, I can break this down into separate terms.First, let me separate the expression:M*(3u - v + 2w) = M*(3u) - M*v + M*(2w)Now, I also remember that scalar multiplication can be factored out of the matrix multiplication. So, M*(3u) is the same as 3*(M*u), and similarly, M*(2w) is 2*(M*w). That should make things easier because I already know what M*u, M*v, and M*w are.So, substituting the known values:= 3*(M*u) - M*v + 2*(M*w)Now plug in the vectors:= 3*[3, -1]^T - [-2, 4]^T + 2*[5, -3]^TOkay, let's compute each scalar multiplication first.Starting with 3*[3, -1]^T:3*3 = 93*(-1) = -3So, that gives [9, -3]^T.Next, 2*[5, -3]^T:2*5 = 102*(-3) = -6So, that gives [10, -6]^T.Now, let's write all the terms with these computed vectors:= [9, -3]^T - [-2, 4]^T + [10, -6]^TNow, I need to perform the vector subtraction and addition. Let's handle the subtraction first: [9, -3]^T - [-2, 4]^T.Subtracting vectors is done component-wise. So, subtract each corresponding component:First component: 9 - (-2) = 9 + 2 = 11Second component: -3 - 4 = -7So, that gives [11, -7]^T.Now, add the last vector [10, -6]^T to this result:[11, -7]^T + [10, -6]^TAgain, component-wise addition:First component: 11 + 10 = 21Second component: -7 + (-6) = -13So, putting it all together, the result is [21, -13]^T.Wait, let me double-check my steps to make sure I didn't make a mistake.First, I distributed M correctly over the linear combination, which is allowed because matrix multiplication is linear. Then, I correctly factored out the scalars 3 and 2, multiplying them with the respective vectors M*u and M*w. Calculating 3*[3, -1]^T gave me [9, -3]^T, which seems right. Then, 2*[5, -3]^T gave me [10, -6]^T, which also looks correct.Subtracting [-2, 4]^T from [9, -3]^T: 9 - (-2) is 11, and -3 - 4 is -7. That seems correct.Adding [10, -6]^T to [11, -7]^T: 11 + 10 is 21, and -7 + (-6) is -13. Yep, that adds up.So, I think my final answer is [21, -13]^T. I don't see any mistakes in my calculations.

💡Okay, so I have this problem to prove that a convex polygon has a center of symmetry if and only if it can be represented as a sum of several line segments. Hmm, that sounds a bit abstract, but let's try to break it down.First, I need to understand what a center of symmetry means. I think it means that there's a point in the polygon such that for every point on the polygon, there's another point directly opposite it through that center. So, if I pick any vertex, there's another vertex that's like its mirror image with respect to the center. That makes sense.Now, the polygon is convex, which means all its interior angles are less than 180 degrees, and every line segment between two vertices stays inside the polygon. So, no dents or anything; it's a smooth shape.The problem says that such a polygon has a center of symmetry if and only if it can be represented as a sum of several line segments. I'm not entirely sure what "sum of several line segments" means in this context. Maybe it means that the polygon can be constructed by adding up or combining line segments in some way? Or perhaps it's referring to the Minkowski sum? I think Minkowski sum is a concept where you add sets by taking all possible sums of their points. So, if I have two line segments, their Minkowski sum would be a parallelogram, right?Wait, so if I have multiple line segments, their Minkowski sum could create more complex shapes. Maybe the polygon can be expressed as the Minkowski sum of several line segments. That would make sense because each line segment contributes to the overall shape.Let me think about the "if and only if" part. That means I need to prove two directions: 1. If a convex polygon has a center of symmetry, then it can be represented as a sum of several line segments.2. If a convex polygon can be represented as a sum of several line segments, then it has a center of symmetry.Okay, starting with the first direction. Suppose the polygon has a center of symmetry, O. For every vertex A, there's another vertex A' such that O is the midpoint of AA'. So, the polygon is symmetric about O.How can I represent this polygon as a sum of line segments? Maybe I can consider each pair of opposite vertices as endpoints of a line segment, and then sum these segments somehow. But I'm not sure how exactly to do that.Alternatively, maybe I can think of the polygon as being generated by translating and scaling line segments around the center O. If I have a set of line segments whose midpoints are all at O, then their Minkowski sum would naturally be symmetric about O. So, if I can express the polygon as such a sum, it would have a center of symmetry.Wait, that seems promising. If I have several line segments, each symmetric about O, then their sum would also be symmetric about O. So, maybe the polygon can be decomposed into these symmetric line segments.But how do I ensure that the sum of these segments reconstructs the original polygon? I think I need to consider the directions and lengths of the segments. Each segment should be aligned in a way that their combination gives the edges of the polygon.Hmm, maybe I need to parameterize the polygon in terms of these segments. Let's say each edge of the polygon can be represented as a scaled and rotated version of a base segment. Then, the sum of all these scaled and rotated segments would give the polygon.But I'm not sure if that's the right approach. Maybe I should look at specific examples. Let's take a simple symmetric convex polygon, like a regular hexagon. It has a center of symmetry. Can I represent it as a sum of line segments?A regular hexagon can be thought of as the Minkowski sum of three line segments along the x, y, and z axes, but scaled appropriately. Wait, no, that might not be exactly right. Actually, a regular hexagon can be constructed by adding three line segments at 60-degree angles to each other. So, each pair of opposite edges is a line segment, and their sum gives the hexagon.Okay, so in this case, the regular hexagon is the Minkowski sum of three line segments. Each segment corresponds to a pair of opposite edges. That makes sense because the hexagon is symmetric, and each segment is symmetric about the center.So, maybe in general, for any convex polygon with a center of symmetry, we can pair up its opposite edges and represent each pair as a line segment, then take the Minkowski sum of these segments to reconstruct the polygon.But wait, not all convex polygons with a center of symmetry are regular. For example, a rectangle is also symmetric, but it's not regular unless it's a square. So, can a rectangle be represented as a sum of line segments?Yes, a rectangle can be seen as the Minkowski sum of two line segments: one along the x-axis and one along the y-axis. Each segment corresponds to the length and width of the rectangle. Their sum gives the rectangle.So, it seems like for both regular and irregular convex polygons with a center of symmetry, we can represent them as Minkowski sums of line segments. Each pair of opposite edges corresponds to a line segment, and their sum reconstructs the polygon.Now, for the converse: if a convex polygon can be represented as a sum of several line segments, does it necessarily have a center of symmetry?Well, if the polygon is the Minkowski sum of line segments, then each segment contributes to the overall shape. If each segment is symmetric about a common center, then their sum should also be symmetric about that center. So, if all the line segments are symmetric about a point O, then the resulting polygon will also be symmetric about O.But what if the line segments are not symmetric about a common center? Then, their sum might not have a center of symmetry. So, perhaps the condition is that the line segments must be symmetric about a common point for their sum to have a center of symmetry.Wait, but the problem statement just says "a sum of several line segments," without specifying anything about their symmetry. So, maybe I need to ensure that the line segments are chosen such that their sum is symmetric.Alternatively, perhaps the way the segments are summed inherently requires them to be symmetric. For example, if you take the Minkowski sum of segments that are not symmetric, you might end up with an asymmetric shape. So, to get a symmetric polygon, the segments must be arranged symmetrically.Therefore, if a convex polygon can be represented as a sum of several line segments, and those segments are arranged symmetrically about a common center, then the polygon has a center of symmetry.Putting it all together, I think the key idea is that a convex polygon has a center of symmetry if and only if it can be expressed as the Minkowski sum of line segments that are symmetric about that center. Each pair of opposite edges of the polygon corresponds to a line segment, and their sum reconstructs the polygon while preserving the symmetry.I need to formalize this idea. Maybe I can consider each edge of the polygon as a vector, and since the polygon is symmetric, each edge has an opposite edge that is its mirror image. Then, the sum of these vectors (or segments) would give the polygon, and the symmetry is preserved because each segment is paired with its opposite.So, in mathematical terms, if the polygon has a center of symmetry O, then for each vertex A_i, there exists a vertex A_{i+n} such that O is the midpoint of A_iA_{i+n}. Then, the vector from O to A_i is the negative of the vector from O to A_{i+n}. Therefore, the polygon can be represented as the sum of these vectors, which are essentially line segments from O to each vertex.But wait, that might not exactly be the case because the edges are between consecutive vertices, not from the center. So, maybe I need to think differently.Perhaps I should consider the edges as vectors and note that for a symmetric polygon, each edge has an opposite edge that is its mirror image. So, the sum of these edge vectors would cancel out in some way, but I'm not sure how that relates to the polygon being a sum of line segments.Alternatively, maybe I can think of the polygon as the convex hull of the sum of line segments. If each line segment is symmetric about O, then their convex hull would also be symmetric about O.I think I'm getting closer. Let me try to outline the proof step by step.First, assume the polygon has a center of symmetry O. Then, for each vertex A_i, there exists a vertex A'_i such that O is the midpoint of A_iA'_i. Now, consider the vectors from O to each A_i. Since the polygon is convex, these vectors can be arranged in order around O.If I take each pair of opposite vectors (A_i and A'_i), they form a line segment symmetric about O. So, the polygon can be thought of as the Minkowski sum of these symmetric line segments.Conversely, if the polygon can be represented as a sum of several line segments, and each segment is symmetric about a common point O, then the resulting polygon is symmetric about O.Therefore, the polygon has a center of symmetry if and only if it can be represented as a sum of several line segments, each symmetric about that center.I think that makes sense. Each line segment contributes to the overall shape, and their symmetry ensures the polygon's symmetry. Conversely, the polygon's symmetry allows it to be decomposed into symmetric line segments whose sum reconstructs the polygon.I should probably write this more formally, using mathematical notation and definitions, but I think the core idea is there.

💡Alright, so I've got this problem about an equilateral triangle ABC. Part a asks for the locus of points X such that AX² = BX² + CX². Hmm, okay, so I need to find all points X where the square of the distance from A to X is equal to the sum of the squares of the distances from B to X and C to X.First, since ABC is equilateral, all sides are equal, and all angles are 60 degrees. Maybe I can use some coordinate geometry to solve this. Let me place the triangle in a coordinate system to make things easier.Let's set point A at (0, 0), point B at (1, 0), and point C at (0.5, √3/2). That way, the triangle is centered nicely in the plane. Now, let X be any point (x, y). Then, AX² is x² + y², BX² is (x - 1)² + y², and CX² is (x - 0.5)² + (y - √3/2)².So, the equation AX² = BX² + CX² becomes:x² + y² = [(x - 1)² + y²] + [(x - 0.5)² + (y - √3/2)²]Let me expand the right-hand side:First term: (x - 1)² + y² = x² - 2x + 1 + y²Second term: (x - 0.5)² + (y - √3/2)² = x² - x + 0.25 + y² - √3 y + 3/4Adding these together:(x² - 2x + 1 + y²) + (x² - x + 0.25 + y² - √3 y + 3/4) = 2x² - 3x + 2y² - √3 y + 2So, the equation becomes:x² + y² = 2x² - 3x + 2y² - √3 y + 2Let me bring everything to one side:0 = x² - 3x + y² - √3 y + 2Hmm, this is a quadratic equation in x and y. Maybe I can complete the squares to find the locus.For the x terms: x² - 3x. Completing the square:x² - 3x = (x - 1.5)² - (1.5)² = (x - 1.5)² - 2.25For the y terms: y² - √3 y. Completing the square:y² - √3 y = (y - (√3)/2)² - ( (√3)/2 )² = (y - (√3)/2)² - 3/4Substituting back into the equation:0 = (x - 1.5)² - 2.25 + (y - (√3)/2)² - 3/4 + 2Simplify constants:-2.25 - 3/4 + 2 = -2.25 - 0.75 + 2 = -3 + 2 = -1So, the equation becomes:(x - 1.5)² + (y - (√3)/2)² - 1 = 0Which simplifies to:(x - 1.5)² + (y - (√3)/2)² = 1So, this is a circle centered at (1.5, √3/2) with radius 1.Wait a minute, but in my coordinate system, point C is at (0.5, √3/2). So, the center of this circle is at (1.5, √3/2), which is symmetric to point C with respect to the midpoint of BC.Is that correct? Let me think. The midpoint of BC is at ((1 + 0.5)/2, (0 + √3/2)/2) = (0.75, √3/4). So, reflecting point C over this midpoint would give (1.5, √3/2), which is indeed the center of the circle.So, the locus is a circle centered at the reflection of C over the midpoint of BC, with radius 1. Since the side length of the triangle is 1, the radius is equal to the side length.Okay, that seems reasonable. So, for part a, the locus is a circle with radius equal to the side length of the equilateral triangle, centered at the reflection of one vertex over the midpoint of the opposite side.Now, moving on to part b: Prove that for the points on the mentioned locus, the pedal triangle with respect to triangle ABC is right-angled.Hmm, the pedal triangle. The pedal triangle of a point X with respect to triangle ABC is formed by projecting X onto the sides of ABC. So, if I drop perpendiculars from X to each side of ABC, the feet of these perpendiculars form the pedal triangle.I need to show that this pedal triangle is right-angled.Given that X lies on the circle found in part a, which is centered at the reflection of C over the midpoint of BC, with radius equal to the side length.Maybe I can use some properties of pedal triangles and the given condition AX² = BX² + CX².Alternatively, perhaps I can use coordinates again.Let me consider the coordinates of X on the circle (x - 1.5)² + (y - √3/2)² = 1.Let me parameterize X on this circle. Let me set:x = 1.5 + cosθy = √3/2 + sinθSo, X is (1.5 + cosθ, √3/2 + sinθ)Now, I need to find the pedal triangle of X with respect to ABC.The pedal triangle is formed by the feet of the perpendiculars from X to the sides of ABC.So, let's find the equations of the sides of ABC and then find the feet.First, side BC: from (1, 0) to (0.5, √3/2). The equation of BC can be found.The slope of BC is (√3/2 - 0)/(0.5 - 1) = (√3/2)/(-0.5) = -√3So, the equation is y - 0 = -√3(x - 1), which simplifies to y = -√3 x + √3Similarly, side AC: from (0, 0) to (0.5, √3/2). The slope is (√3/2 - 0)/(0.5 - 0) = √3Equation: y = √3 xSide AB: from (0, 0) to (1, 0). It's the x-axis, y = 0.Now, find the feet of the perpendiculars from X to each side.Let's denote the feet as D, E, F on BC, AC, AB respectively.First, find foot D on BC.The line BC is y = -√3 x + √3. The slope is -√3, so the perpendicular has slope 1/√3.Equation of perpendicular from X: y - y_X = (1/√3)(x - x_X)So, y = (1/√3)x + (y_X - (1/√3)x_X)Find intersection with BC:(1/√3)x + (y_X - (1/√3)x_X) = -√3 x + √3Multiply both sides by √3 to eliminate denominators:x + √3(y_X - (1/√3)x_X) = -3x + 3Bring all terms to left:x + √3 y_X - x_X + 3x - 3 = 0Combine like terms:4x + √3 y_X - x_X - 3 = 0Solve for x:4x = 3 + x_X - √3 y_Xx = (3 + x_X - √3 y_X)/4Then, y = -√3 x + √3Substitute x:y = -√3*(3 + x_X - √3 y_X)/4 + √3Simplify:y = (-3√3 - √3 x_X + 3 y_X)/4 + √3Combine terms:y = (-3√3 - √3 x_X + 3 y_X + 4√3)/4Simplify:y = (√3 - √3 x_X + 3 y_X)/4So, coordinates of D are:D = ( (3 + x_X - √3 y_X)/4 , (√3 - √3 x_X + 3 y_X)/4 )Similarly, find foot E on AC.Line AC: y = √3 xSlope is √3, so perpendicular slope is -1/√3Equation of perpendicular from X: y - y_X = (-1/√3)(x - x_X)Find intersection with AC:(-1/√3)x + (y_X + (1/√3)x_X) = √3 xMultiply both sides by √3:- x + √3 y_X + x_X = 3xBring terms together:- x - 3x + √3 y_X + x_X = 0-4x + √3 y_X + x_X = 0Solve for x:4x = √3 y_X + x_Xx = (√3 y_X + x_X)/4Then, y = √3 x = √3*(√3 y_X + x_X)/4 = (3 y_X + √3 x_X)/4So, coordinates of E are:E = ( (√3 y_X + x_X)/4 , (3 y_X + √3 x_X)/4 )Finally, foot F on AB, which is y = 0.The perpendicular from X to AB is vertical, since AB is horizontal.So, F has the same x-coordinate as X, and y = 0.Thus, F = (x_X, 0)Now, we have the three feet D, E, F.To find the pedal triangle, we need to find the triangle DEF.To show that DEF is right-angled, we can compute the vectors or the slopes and see if any two sides are perpendicular.Alternatively, compute the distances and use the Pythagorean theorem.Let me compute the coordinates of D, E, F in terms of x_X and y_X.Given X = (1.5 + cosθ, √3/2 + sinθ)So, x_X = 1.5 + cosθy_X = √3/2 + sinθLet me compute coordinates of D:D_x = (3 + x_X - √3 y_X)/4= (3 + 1.5 + cosθ - √3*(√3/2 + sinθ))/4Simplify:= (4.5 + cosθ - (3/2 + √3 sinθ))/4= (4.5 - 1.5 + cosθ - √3 sinθ)/4= (3 + cosθ - √3 sinθ)/4Similarly, D_y = (√3 - √3 x_X + 3 y_X)/4= (√3 - √3*(1.5 + cosθ) + 3*(√3/2 + sinθ))/4Simplify:= (√3 - 1.5√3 - √3 cosθ + (3√3)/2 + 3 sinθ)/4Combine like terms:= (√3 - 1.5√3 + 1.5√3 - √3 cosθ + 3 sinθ)/4= (√3 - √3 cosθ + 3 sinθ)/4So, D = ( (3 + cosθ - √3 sinθ)/4 , (√3 - √3 cosθ + 3 sinθ)/4 )Similarly, compute E:E_x = (√3 y_X + x_X)/4= (√3*(√3/2 + sinθ) + 1.5 + cosθ)/4Simplify:= ( (3/2 + √3 sinθ) + 1.5 + cosθ ) /4= (3/2 + 1.5 + √3 sinθ + cosθ)/4= (3 + √3 sinθ + cosθ)/4E_y = (3 y_X + √3 x_X)/4= (3*(√3/2 + sinθ) + √3*(1.5 + cosθ))/4Simplify:= ( (3√3)/2 + 3 sinθ + (3√3)/2 + √3 cosθ ) /4Combine like terms:= (3√3 + 3 sinθ + √3 cosθ)/4So, E = ( (3 + √3 sinθ + cosθ)/4 , (3√3 + 3 sinθ + √3 cosθ)/4 )And F = (x_X, 0) = (1.5 + cosθ, 0)Now, we have coordinates for D, E, F.To check if triangle DEF is right-angled, let's compute the vectors DE, DF, and EF and see if any two are perpendicular.First, compute DE:E - D = ( (3 + √3 sinθ + cosθ)/4 - (3 + cosθ - √3 sinθ)/4 , (3√3 + 3 sinθ + √3 cosθ)/4 - (√3 - √3 cosθ + 3 sinθ)/4 )Simplify x-coordinate:(3 + √3 sinθ + cosθ - 3 - cosθ + √3 sinθ)/4 = (2√3 sinθ)/4 = (√3 sinθ)/2Y-coordinate:(3√3 + 3 sinθ + √3 cosθ - √3 + √3 cosθ - 3 sinθ)/4Simplify:(3√3 - √3 + 3 sinθ - 3 sinθ + √3 cosθ + √3 cosθ)/4= (2√3 + 2√3 cosθ)/4 = (√3(2 + 2 cosθ))/4 = (√3(1 + cosθ))/2So, DE vector is ( (√3 sinθ)/2 , (√3(1 + cosθ))/2 )Similarly, compute DF:F - D = (1.5 + cosθ - (3 + cosθ - √3 sinθ)/4 , 0 - (√3 - √3 cosθ + 3 sinθ)/4 )Simplify x-coordinate:(6 + 4 cosθ - 3 - cosθ + √3 sinθ)/4 = (3 + 3 cosθ + √3 sinθ)/4Y-coordinate:( -√3 + √3 cosθ - 3 sinθ ) /4So, DF vector is ( (3 + 3 cosθ + √3 sinθ)/4 , (-√3 + √3 cosθ - 3 sinθ)/4 )Similarly, compute EF:F - E = (1.5 + cosθ - (3 + √3 sinθ + cosθ)/4 , 0 - (3√3 + 3 sinθ + √3 cosθ)/4 )Simplify x-coordinate:(6 + 4 cosθ - 3 - √3 sinθ - cosθ)/4 = (3 + 3 cosθ - √3 sinθ)/4Y-coordinate:( -3√3 - 3 sinθ - √3 cosθ ) /4So, EF vector is ( (3 + 3 cosθ - √3 sinθ)/4 , (-3√3 - 3 sinθ - √3 cosθ)/4 )Now, to check if any two vectors are perpendicular, their dot product should be zero.Let's check DE and DF:Dot product = ( (√3 sinθ)/2 )*( (3 + 3 cosθ + √3 sinθ)/4 ) + ( (√3(1 + cosθ))/2 )*( (-√3 + √3 cosθ - 3 sinθ)/4 )Simplify:= [ (√3 sinθ)(3 + 3 cosθ + √3 sinθ) + (√3(1 + cosθ))(-√3 + √3 cosθ - 3 sinθ) ] / 8Let me compute each term:First term: (√3 sinθ)(3 + 3 cosθ + √3 sinθ)= 3√3 sinθ + 3√3 sinθ cosθ + 3 sin²θSecond term: (√3(1 + cosθ))(-√3 + √3 cosθ - 3 sinθ)= √3*(-√3)(1 + cosθ) + √3*(√3 cosθ)(1 + cosθ) + √3*(-3 sinθ)(1 + cosθ)= -3(1 + cosθ) + 3 cosθ(1 + cosθ) - 3√3 sinθ(1 + cosθ)Simplify:= -3 - 3 cosθ + 3 cosθ + 3 cos²θ - 3√3 sinθ - 3√3 sinθ cosθCombine like terms:= -3 + 3 cos²θ - 3√3 sinθ - 3√3 sinθ cosθNow, add the first and second terms:First term: 3√3 sinθ + 3√3 sinθ cosθ + 3 sin²θSecond term: -3 + 3 cos²θ - 3√3 sinθ - 3√3 sinθ cosθAdding together:3√3 sinθ - 3√3 sinθ + 3√3 sinθ cosθ - 3√3 sinθ cosθ + 3 sin²θ - 3 + 3 cos²θSimplify:0 + 0 + 3 sin²θ - 3 + 3 cos²θ= 3(sin²θ + cos²θ) - 3= 3(1) - 3 = 0So, the dot product is zero. Therefore, DE and DF are perpendicular.Thus, triangle DEF has a right angle at D.Therefore, the pedal triangle is right-angled.**Final Answer**a) The locus is a circle with radius equal to the side length of the equilateral triangle, centered at the reflection of one vertex over the midpoint of the opposite side. b) The pedal triangle is right-angled.boxed{text{a circle with radius equal to the side length of the equilateral triangle, centered at the reflection of one vertex over the midpoint of the opposite side}}boxed{text{The pedal triangle is right-angled.}}

💡Alright, so I have this problem where I need to find the number of maps from set A to set B. Let me make sure I understand what a "map" means in this context. From what I remember, a map, or function, from set A to set B assigns each element in A to exactly one element in B. So, for every element in A, I have to choose where it goes in B.Okay, set A is given as {D, U, K, E}, which has 4 elements. Set B is {M, A, T, H}, which also has 4 elements. So, each of the 4 elements in A can be mapped to any of the 4 elements in B. That means for each element in A, there are 4 choices in B.Let me think about how this works. If I have the first element D in A, I can map it to M, A, T, or H. That's 4 options. Then, for the next element U, I again have 4 options in B, regardless of where D was mapped. The same goes for K and E. So, for each of the 4 elements in A, there are 4 choices in B.To find the total number of maps, I need to multiply the number of choices for each element together. That would be 4 (for D) multiplied by 4 (for U) multiplied by 4 (for K) multiplied by 4 (for E). So, 4 × 4 × 4 × 4.Let me calculate that. 4 × 4 is 16. 16 × 4 is 64. 64 × 4 is 256. So, there are 256 different maps from A to B.Wait, let me double-check. Is there any restriction or something I'm missing? The problem just asks for the number of maps, and I don't see any restrictions mentioned, like injective or surjective functions. So, it's just the total number of functions, which is indeed 4^4, which is 256.I think that makes sense. Each element in A has independent choices in B, and since there are 4 elements, it's 4 raised to the power of 4. Yeah, I'm pretty confident the answer is 256.