answer：Okay, so I have this problem about a regular octagon, ABCDEFGH. The sides AB and GH are extended to meet at point Q, and I need to find the measure of angle Q. Hmm, okay, let me think about how to approach this.First, I remember that in a regular octagon, all sides are equal and all interior angles are equal. I think the measure of each interior angle in a regular octagon is 135 degrees. Let me verify that. The formula for the sum of interior angles of an n-sided polygon is 180(n-2) degrees. So for an octagon, n=8, so the sum is 180*(8-2) = 1080 degrees. Therefore, each interior angle is 1080/8 = 135 degrees. Yep, that checks out.Now, the problem says sides AB and GH are extended to meet at point Q. I need to visualize this. Let me try to sketch a regular octagon and label the vertices A, B, C, D, E, F, G, H in order. So, AB is one side, and GH is another side. If I extend AB beyond B and GH beyond H, they should meet at some point Q outside the octagon.Wait, actually, the problem says "extended to meet at point Q." So, maybe AB is extended beyond A and GH is extended beyond G? Hmm, I need to clarify that. Let me think: in a regular octagon, each side is connected to the next vertex. So, AB is connected to B, which is connected to C, and so on. GH is connected to H, which is connected to A. So, if I extend AB beyond A and GH beyond H, they might meet at a point Q. Alternatively, if I extend AB beyond B and GH beyond G, they might meet at another point. I need to figure out which extension leads to the meeting point Q.Wait, the problem doesn't specify which direction to extend, just says "extended to meet at point Q." So, perhaps it's the extensions beyond A and beyond H? Let me try to imagine that. If I extend AB beyond A, it's going away from the octagon, and similarly, GH beyond H is also going away from the octagon. Depending on the octagon's orientation, these two extensions might meet at some external point Q.Alternatively, if I extend AB beyond B and GH beyond G, they might meet at another external point. Hmm, I think it's more likely that the extensions beyond A and beyond H meet because those sides are adjacent in the sequence, but I'm not entirely sure. Maybe I should consider both possibilities.Wait, actually, in a regular octagon, each side is separated by 45-degree angles because the central angles are 360/8 = 45 degrees. So, the sides are spaced 45 degrees apart. So, if I think of the octagon as being inscribed in a circle, each vertex is 45 degrees apart on the circumference.If I consider sides AB and GH, how many sides apart are they? Let's count: A, B, C, D, E, F, G, H. So, from AB to GH, that's moving from A to G, which is 6 sides apart? Wait, no, AB is the side between A and B, and GH is the side between G and H. So, from AB to GH, how many sides apart are they? Let's see: AB is side 1, BC is side 2, CD is side 3, DE is side 4, EF is side 5, FG is side 6, GH is side 7. So, AB is side 1 and GH is side 7, so they are 6 sides apart? Wait, that doesn't seem right because in an octagon, sides are adjacent or opposite.Wait, actually, in an octagon, each side has a corresponding opposite side. So, AB is opposite to EF, BC is opposite to FG, and so on. So, GH is opposite to which side? Let me see: starting from A, the opposite side would be E, so AB is opposite to EF, BC opposite to FG, CD opposite to GH. So, CD is opposite to GH. Therefore, AB and GH are not opposite sides; they are separated by some number of sides.Wait, maybe I should think in terms of the number of edges between AB and GH. Starting from AB, the next side is BC, then CD, then DE, then EF, then FG, then GH. So, from AB to GH, it's 6 sides apart? That seems like a lot. Alternatively, maybe it's 2 sides apart because after AB comes BC, then CD, then DE, then EF, then FG, then GH. So, from AB to GH is 6 sides, but in an octagon, the maximum distance between sides is 4, because beyond that, it's shorter the other way.Wait, maybe I'm overcomplicating this. Let me think about the positions of AB and GH. If I label the octagon in order, A, B, C, D, E, F, G, H, then AB is between A and B, and GH is between G and H. So, in terms of the circle, AB is at some angle, and GH is at another angle. The angle between AB and GH would be the central angle between A and G, which is 6*45 = 270 degrees? Wait, no, because each side corresponds to a central angle of 45 degrees. So, from A to G, that's 6 steps, each of 45 degrees, so 6*45 = 270 degrees. But since a circle is 360 degrees, 270 degrees is three-quarters of the circle. So, the angle between AB and GH is 270 degrees? That seems too much.Wait, maybe I should think about the external angles. When you extend sides AB and GH, the angle at point Q is formed by these two extensions. So, perhaps I need to find the angle between the extensions of AB and GH beyond the octagon.Let me try to draw this mentally. If I have a regular octagon, and I extend AB beyond A and GH beyond H, these two lines will meet at some point Q outside the octagon. The angle at Q is the angle between these two extended sides.Alternatively, if I extend AB beyond B and GH beyond G, they might meet at another point. But the problem says "extended to meet at point Q," so I think it's the extensions beyond A and beyond H that meet at Q.Wait, actually, in a regular octagon, the sides AB and GH are not adjacent. AB is between A and B, and GH is between G and H. So, in the sequence, after H comes A again. So, GH is connected to H, which is connected back to A. So, if I extend AB beyond A, it's going in the direction away from B, and if I extend GH beyond H, it's going in the direction away from G. So, these two extensions might meet at a point Q outside the octagon.Okay, so I need to find the measure of angle Q, which is the angle between the extensions of AB and GH beyond A and H, respectively.To find this angle, maybe I can use some properties of regular octagons and triangles. Let me think about the coordinates of the octagon. If I place the octagon on a coordinate system, with center at the origin, and one vertex at (1,0), then the coordinates of the vertices can be determined using the central angles.But maybe that's too complicated. Alternatively, I can consider the angles formed by the sides and their extensions.Since each interior angle of the octagon is 135 degrees, the exterior angle is 180 - 135 = 45 degrees. So, each exterior angle is 45 degrees.When we extend a side of the octagon, the angle outside the octagon at that vertex is equal to the exterior angle, which is 45 degrees.So, if I extend AB beyond A, the angle between AB and the extension beyond A is 45 degrees. Similarly, if I extend GH beyond H, the angle between GH and the extension beyond H is 45 degrees.Wait, but how does this help me find angle Q?Maybe I can consider triangle AQG, where Q is the intersection point of the extensions of AB and GH. In this triangle, I can find the angles at A and G, and then use the fact that the sum of angles in a triangle is 180 degrees to find angle Q.But first, I need to figure out the angles at A and G in triangle AQG.At point A, the original angle of the octagon is 135 degrees. When we extend AB beyond A, the angle between AB and the extension is 45 degrees, as I thought earlier. Similarly, at point G, the original angle is 135 degrees, and extending GH beyond H creates another 45-degree angle.Wait, but in triangle AQG, the angles at A and G are not the same as the exterior angles. Let me think carefully.When we extend AB beyond A, the direction of the extension is such that it forms a straight line with AB. Similarly, extending GH beyond H forms a straight line with GH. The angle at Q is formed by these two extensions.So, perhaps I can think of the lines AQ and GQ as the extensions of AB and GH, respectively. Then, the angle at Q is the angle between these two lines.To find this angle, I might need to know the angle between the original sides AB and GH, and then see how extending them affects the angle.Wait, in a regular octagon, the sides are equally spaced around the center. So, the angle between two adjacent sides is 45 degrees. But AB and GH are not adjacent; they are separated by several sides.Let me count the number of sides between AB and GH. Starting from AB, the next side is BC, then CD, DE, EF, FG, and then GH. So, from AB to GH, there are 6 sides in between. But in an octagon, moving from AB to GH is actually moving 6 steps forward, but since it's a circle, moving 6 steps forward is the same as moving 2 steps backward (because 8 - 6 = 2). So, the angle between AB and GH is 2 * 45 = 90 degrees.Wait, is that correct? Let me think. Each side corresponds to a central angle of 45 degrees. So, the angle between AB and GH would be the central angle between points A and G. Since A is the first vertex and G is the seventh vertex, the central angle between A and G is (7 - 1) * 45 = 6 * 45 = 270 degrees. But since a circle is 360 degrees, the smaller angle between A and G is 360 - 270 = 90 degrees. So, the central angle between A and G is 90 degrees.Therefore, the angle between the sides AB and GH is 90 degrees. But how does this relate to the angle at Q?Wait, when we extend AB and GH, the angle at Q is related to the angle between these two sides. Since the central angle is 90 degrees, the angle between the extensions might be related.But I'm not sure if it's exactly 90 degrees. Let me think about the triangle formed by points A, G, and Q.In triangle AQG, we have points A and G on the octagon, and Q outside. The sides AQ and GQ are the extensions of AB and GH, respectively. So, the angle at Q is the angle between AQ and GQ.To find this angle, I might need to use the Law of Sines or Cosines if I can find the lengths of the sides or other angles.Alternatively, maybe I can consider the angles at A and G in triangle AQG.At point A, the original angle of the octagon is 135 degrees. When we extend AB beyond A, the angle between AB and AQ is 180 - 135 = 45 degrees. Similarly, at point G, the original angle is 135 degrees, and the angle between GH and GQ is 180 - 135 = 45 degrees.Wait, so in triangle AQG, the angles at A and G are each 45 degrees. Therefore, the sum of angles in triangle AQG is 180 degrees, so angle Q is 180 - 45 - 45 = 90 degrees.Oh, so angle Q is 90 degrees. That makes sense.Let me recap to make sure I didn't make a mistake. Each interior angle of the octagon is 135 degrees. When we extend a side, the external angle is 45 degrees. So, at points A and G, the angles between the original sides and the extensions are each 45 degrees. In triangle AQG, these two angles are 45 degrees each, so the remaining angle at Q is 90 degrees.Yes, that seems correct. So, the measure of angle Q is 90 degrees.

answer：Okay, so I need to find the value of (sin left( sin^{-1} frac{3}{5} + tan^{-1} left(frac{1}{2}right) right)). Hmm, that looks a bit complicated, but I think I can break it down step by step.First, let me understand what each part means. The expression inside the sine function is the sum of two inverse trigonometric functions: (sin^{-1} frac{3}{5}) and (tan^{-1} frac{1}{2}). So, I can think of this as (sin(a + b)), where (a = sin^{-1} frac{3}{5}) and (b = tan^{-1} frac{1}{2}).Alright, so I need to find (sin(a + b)). I remember there's a formula for the sine of a sum of two angles: (sin(a + b) = sin a cos b + cos a sin b). That seems right. So, if I can find (sin a), (cos a), (sin b), and (cos b), I can plug them into this formula.Let me start with (a = sin^{-1} frac{3}{5}). That means (sin a = frac{3}{5}). Since (sin^2 a + cos^2 a = 1), I can find (cos a) by rearranging this identity. So, (cos a = sqrt{1 - sin^2 a}). Plugging in the value, that would be (sqrt{1 - left(frac{3}{5}right)^2}). Let me compute that:[sqrt{1 - frac{9}{25}} = sqrt{frac{25}{25} - frac{9}{25}} = sqrt{frac{16}{25}} = frac{4}{5}]Okay, so (cos a = frac{4}{5}). That wasn't too bad.Now, moving on to (b = tan^{-1} frac{1}{2}). That means (tan b = frac{1}{2}). I need to find (sin b) and (cos b). I remember that (tan b = frac{sin b}{cos b}), so if (tan b = frac{1}{2}), I can think of a right triangle where the opposite side is 1 and the adjacent side is 2. Then, the hypotenuse would be (sqrt{1^2 + 2^2} = sqrt{1 + 4} = sqrt{5}).So, in this triangle, (sin b = frac{text{opposite}}{text{hypotenuse}} = frac{1}{sqrt{5}}) and (cos b = frac{text{adjacent}}{text{hypotenuse}} = frac{2}{sqrt{5}}). Let me write that down:[sin b = frac{1}{sqrt{5}}, quad cos b = frac{2}{sqrt{5}}]Wait, is that right? Let me double-check. If (tan b = frac{1}{2}), then yes, opposite is 1, adjacent is 2, hypotenuse is (sqrt{5}). So, (sin b = frac{1}{sqrt{5}}) and (cos b = frac{2}{sqrt{5}}). Okay, that seems correct.Now, going back to the angle addition formula:[sin(a + b) = sin a cos b + cos a sin b]I already have all the values:- (sin a = frac{3}{5})- (cos a = frac{4}{5})- (sin b = frac{1}{sqrt{5}})- (cos b = frac{2}{sqrt{5}})Let me plug these into the formula:[sin(a + b) = left(frac{3}{5}right) left(frac{2}{sqrt{5}}right) + left(frac{4}{5}right) left(frac{1}{sqrt{5}}right)]Calculating each term separately:First term: (frac{3}{5} times frac{2}{sqrt{5}} = frac{6}{5sqrt{5}})Second term: (frac{4}{5} times frac{1}{sqrt{5}} = frac{4}{5sqrt{5}})Adding them together:[frac{6}{5sqrt{5}} + frac{4}{5sqrt{5}} = frac{6 + 4}{5sqrt{5}} = frac{10}{5sqrt{5}} = frac{2}{sqrt{5}}]Hmm, (frac{2}{sqrt{5}}) is the result. But usually, we rationalize the denominator. So, multiplying numerator and denominator by (sqrt{5}):[frac{2}{sqrt{5}} times frac{sqrt{5}}{sqrt{5}} = frac{2sqrt{5}}{5}]So, the final value is (frac{2sqrt{5}}{5}). Let me just make sure I didn't make any mistakes in my calculations.Starting from the beginning: (sin^{-1} frac{3}{5}) gives me (a) where (sin a = frac{3}{5}) and (cos a = frac{4}{5}). Then, (tan^{-1} frac{1}{2}) gives me (b) where (tan b = frac{1}{2}), leading to (sin b = frac{1}{sqrt{5}}) and (cos b = frac{2}{sqrt{5}}). Plugging into the sine addition formula, I get (frac{6}{5sqrt{5}} + frac{4}{5sqrt{5}} = frac{10}{5sqrt{5}} = frac{2}{sqrt{5}}), which simplifies to (frac{2sqrt{5}}{5}). That seems consistent.I think I did everything correctly. I used the angle addition formula, found all the necessary trigonometric values using the Pythagorean identity and the definition of tangent, and then simplified the result. So, I feel confident that (frac{2sqrt{5}}{5}) is the correct answer.**Final Answer**boxed{dfrac{2sqrt{5}}{5}}

answer：Okay, so I'm trying to solve this problem about a triangular prism (ABC - A_1B_1C_1). Let me first visualize the prism. It's a triangular prism, so the base is triangle (ABC) and the top face is triangle (A_1B_1C_1). They mentioned that (AA_1) is perpendicular to the base (ABC), which means the prism is a right prism. Also, (AB) is perpendicular to (AC), so triangle (ABC) is a right-angled triangle at (A). Moreover, (AC = AB = AA_1), so all these edges are equal in length. Let me denote the length of each as (s). So, (AB = AC = AA_1 = s).Points (E) and (F) are midpoints of edges (BC) and (A_1A) respectively. So, (E) is the midpoint of (BC), which is the hypotenuse of the right triangle (ABC). Since (AB = AC = s), triangle (ABC) is an isosceles right-angled triangle, so (BC = ssqrt{2}). Therefore, (BE = EC = frac{ssqrt{2}}{2} = frac{s}{sqrt{2}}).Point (F) is the midpoint of (A_1A). Since (AA_1 = s), then (AF = FA_1 = frac{s}{2}).Point (G) is on edge (CC_1) such that (C_1F) is parallel to the plane (AEG). So, I need to find the position of (G) on (CC_1) such that this condition holds.Let me tackle part 1 first: Find the value of (frac{CG}{CC_1}).Since (C_1F) is parallel to plane (AEG), and (C_1F) is a line, for it to be parallel to the plane, it must be parallel to some line in the plane (AEG). Alternatively, the direction vector of (C_1F) must be perpendicular to the normal vector of plane (AEG). Maybe I can use vectors to solve this.Let me assign coordinates to the points to make it easier. Let me place point (A) at the origin ((0, 0, 0)). Since (AB) is perpendicular to (AC), I can let (AB) lie along the y-axis and (AC) lie along the x-axis. So, point (B) is at ((0, s, 0)), point (C) is at ((s, 0, 0)). Since (AA_1) is perpendicular to the base, it goes along the z-axis, so point (A_1) is at ((0, 0, s)). Then, points (B_1) and (C_1) can be found by translating (B) and (C) along the z-axis by (s). So, (B_1) is at ((0, s, s)) and (C_1) is at ((s, 0, s)).Point (E) is the midpoint of (BC). Coordinates of (B) are ((0, s, 0)) and (C) are ((s, 0, 0)). So, midpoint (E) has coordinates (left(frac{0 + s}{2}, frac{s + 0}{2}, frac{0 + 0}{2}right) = left(frac{s}{2}, frac{s}{2}, 0right)).Point (F) is the midpoint of (A_1A). Coordinates of (A_1) are ((0, 0, s)) and (A) is ((0, 0, 0)). So, midpoint (F) is at (left(frac{0 + 0}{2}, frac{0 + 0}{2}, frac{s + 0}{2}right) = (0, 0, frac{s}{2})).Point (G) is on edge (CC_1). Let me parametrize point (G). Since (CC_1) goes from (C(s, 0, 0)) to (C_1(s, 0, s)), any point on (CC_1) can be written as (C + t(C_1 - C)) where (t) ranges from 0 to 1. So, coordinates of (G) are ((s, 0, 0 + t(s - 0)) = (s, 0, ts)). So, (G = (s, 0, ts)). We need to find the value of (t) such that (C_1F) is parallel to plane (AEG).First, let me find the direction vector of (C_1F). Coordinates of (C_1) are ((s, 0, s)) and (F) is ((0, 0, frac{s}{2})). So, vector (C_1F) is (F - C_1 = (0 - s, 0 - 0, frac{s}{2} - s) = (-s, 0, -frac{s}{2})).Now, for (C_1F) to be parallel to plane (AEG), the direction vector of (C_1F) must be perpendicular to the normal vector of plane (AEG). So, I need to find the normal vector of plane (AEG).To find the normal vector of plane (AEG), I can take two vectors lying on the plane and compute their cross product. Let's take vectors (AE) and (AG).Coordinates of (A) are ((0, 0, 0)), (E) is ((frac{s}{2}, frac{s}{2}, 0)), and (G) is ((s, 0, ts)).Vector (AE = E - A = (frac{s}{2}, frac{s}{2}, 0)).Vector (AG = G - A = (s, 0, ts)).Compute the cross product (AE times AG):Let me denote (AE = (frac{s}{2}, frac{s}{2}, 0)) and (AG = (s, 0, ts)).Cross product is:[begin{vmatrix}mathbf{i} & mathbf{j} & mathbf{k} frac{s}{2} & frac{s}{2} & 0 s & 0 & ts end{vmatrix}]Calculating determinant:[mathbf{i} left( frac{s}{2} cdot ts - 0 cdot 0 right) - mathbf{j} left( frac{s}{2} cdot ts - 0 cdot s right) + mathbf{k} left( frac{s}{2} cdot 0 - frac{s}{2} cdot s right)]Simplify each component:- i-component: (frac{s}{2} cdot ts = frac{s^2 t}{2})- j-component: (-left( frac{s}{2} cdot ts right) = -frac{s^2 t}{2})- k-component: (frac{s}{2} cdot 0 - frac{s}{2} cdot s = -frac{s^2}{2})So, the cross product is:[left( frac{s^2 t}{2}, -frac{s^2 t}{2}, -frac{s^2}{2} right )]This is the normal vector of plane (AEG). Let me denote this as (mathbf{n} = left( frac{s^2 t}{2}, -frac{s^2 t}{2}, -frac{s^2}{2} right )).Now, vector (C_1F) is ((-s, 0, -frac{s}{2})). For (C_1F) to be parallel to plane (AEG), the direction vector (C_1F) must be perpendicular to the normal vector (mathbf{n}). So, their dot product must be zero.Compute the dot product:[(-s) cdot left( frac{s^2 t}{2} right ) + 0 cdot left( -frac{s^2 t}{2} right ) + left( -frac{s}{2} right ) cdot left( -frac{s^2}{2} right ) = 0]Simplify each term:First term: (-s cdot frac{s^2 t}{2} = -frac{s^3 t}{2})Second term: 0Third term: (-frac{s}{2} cdot -frac{s^2}{2} = frac{s^3}{4})So, overall:[-frac{s^3 t}{2} + frac{s^3}{4} = 0]Let me factor out (frac{s^3}{4}):[frac{s^3}{4} (-2t + 1) = 0]Since (s neq 0), we have:[-2t + 1 = 0 implies -2t = -1 implies t = frac{1}{2}]Therefore, point (G) is at (t = frac{1}{2}) along (CC_1). So, the coordinates of (G) are ((s, 0, frac{s}{2})).Therefore, (CG = frac{s}{2}) and (CC_1 = s), so the ratio (frac{CG}{CC_1} = frac{frac{s}{2}}{s} = frac{1}{2}).So, the answer to part 1 is (boxed{dfrac{1}{2}}).Moving on to part 2: Prove that (EG perp A_1C).First, let me find the coordinates of points (E), (G), (A_1), and (C).- (E) is (left(frac{s}{2}, frac{s}{2}, 0right))- (G) is ((s, 0, frac{s}{2}))- (A_1) is ((0, 0, s))- (C) is ((s, 0, 0))So, vector (EG = G - E = (s - frac{s}{2}, 0 - frac{s}{2}, frac{s}{2} - 0) = left( frac{s}{2}, -frac{s}{2}, frac{s}{2} right )).Vector (A_1C = C - A_1 = (s - 0, 0 - 0, 0 - s) = (s, 0, -s)).To prove (EG perp A_1C), their dot product should be zero.Compute the dot product:[left( frac{s}{2} right ) cdot s + left( -frac{s}{2} right ) cdot 0 + left( frac{s}{2} right ) cdot (-s) = frac{s^2}{2} + 0 - frac{s^2}{2} = 0]Since the dot product is zero, vectors (EG) and (A_1C) are perpendicular. Therefore, (EG perp A_1C).So, part 2 is proved.Now, part 3: Find the cosine value of the dihedral angle (A_1 - AG - E).Dihedral angle is the angle between two planes. In this case, the dihedral angle along edge (AG) between the planes (A_1AG) and (EAG).To find the cosine of the dihedral angle, I can find the angle between the normal vectors of the two planes.First, find the normal vectors of the two planes.Plane (A_1AG): Points (A_1), (A), (G).Plane (EAG): Points (E), (A), (G).Let me find the normal vectors for both planes.For plane (A_1AG):Points (A_1(0, 0, s)), (A(0, 0, 0)), (G(s, 0, frac{s}{2})).Vectors in this plane can be (A_1A) and (A_1G).Compute vectors:(A_1A = A - A_1 = (0 - 0, 0 - 0, 0 - s) = (0, 0, -s))(A_1G = G - A_1 = (s - 0, 0 - 0, frac{s}{2} - s) = (s, 0, -frac{s}{2}))Compute the cross product (A_1A times A_1G) to get the normal vector.Cross product:[begin{vmatrix}mathbf{i} & mathbf{j} & mathbf{k} 0 & 0 & -s s & 0 & -frac{s}{2} end{vmatrix}]Calculating determinant:[mathbf{i} (0 cdot -frac{s}{2} - (-s) cdot 0) - mathbf{j} (0 cdot -frac{s}{2} - (-s) cdot s) + mathbf{k} (0 cdot 0 - 0 cdot s)]Simplify each component:- i-component: (0 - 0 = 0)- j-component: (- (0 - (-s^2)) = - (s^2))- k-component: (0 - 0 = 0)So, the normal vector is ((0, -s^2, 0)). We can simplify this to ((0, -1, 0)) by dividing by (s^2).For plane (EAG):Points (Eleft(frac{s}{2}, frac{s}{2}, 0right)), (A(0, 0, 0)), (G(s, 0, frac{s}{2})).Vectors in this plane can be (AE) and (AG).Compute vectors:(AE = E - A = left( frac{s}{2}, frac{s}{2}, 0 right ))(AG = G - A = (s, 0, frac{s}{2}))Compute the cross product (AE times AG) to get the normal vector.We already computed this earlier in part 1:[mathbf{n} = left( frac{s^2 t}{2}, -frac{s^2 t}{2}, -frac{s^2}{2} right )]But wait, in part 1, (t = frac{1}{2}), so substituting (t = frac{1}{2}):[mathbf{n} = left( frac{s^2 cdot frac{1}{2}}{2}, -frac{s^2 cdot frac{1}{2}}{2}, -frac{s^2}{2} right ) = left( frac{s^2}{4}, -frac{s^2}{4}, -frac{s^2}{2} right )]But actually, in part 1, the cross product was computed as:[left( frac{s^2 t}{2}, -frac{s^2 t}{2}, -frac{s^2}{2} right )]But since (t = frac{1}{2}), plugging that in:[left( frac{s^2 cdot frac{1}{2}}{2}, -frac{s^2 cdot frac{1}{2}}{2}, -frac{s^2}{2} right ) = left( frac{s^2}{4}, -frac{s^2}{4}, -frac{s^2}{2} right )]Alternatively, since the normal vector can be scaled, let me compute it again for clarity.Compute cross product (AE times AG):(AE = left( frac{s}{2}, frac{s}{2}, 0 right ))(AG = (s, 0, frac{s}{2}))Cross product:[begin{vmatrix}mathbf{i} & mathbf{j} & mathbf{k} frac{s}{2} & frac{s}{2} & 0 s & 0 & frac{s}{2} end{vmatrix}]Calculating determinant:[mathbf{i} left( frac{s}{2} cdot frac{s}{2} - 0 cdot 0 right ) - mathbf{j} left( frac{s}{2} cdot frac{s}{2} - 0 cdot s right ) + mathbf{k} left( frac{s}{2} cdot 0 - frac{s}{2} cdot s right )]Simplify each component:- i-component: (frac{s^2}{4})- j-component: (-frac{s^2}{4})- k-component: (-frac{s^2}{2})So, the normal vector is (left( frac{s^2}{4}, -frac{s^2}{4}, -frac{s^2}{2} right )). We can factor out (frac{s^2}{4}) to get (left( 1, -1, -2 right )).So, normal vector for plane (EAG) is ((1, -1, -2)).Earlier, for plane (A_1AG), the normal vector was ((0, -1, 0)).Now, to find the cosine of the dihedral angle between the two planes, we can use the dot product of their normal vectors.Let me denote normal vector of plane (A_1AG) as (mathbf{n_1} = (0, -1, 0)) and normal vector of plane (EAG) as (mathbf{n_2} = (1, -1, -2)).The cosine of the angle between the two planes is given by:[cos theta = frac{|mathbf{n_1} cdot mathbf{n_2}|}{|mathbf{n_1}| |mathbf{n_2}|}]Compute the dot product:[mathbf{n_1} cdot mathbf{n_2} = 0 cdot 1 + (-1) cdot (-1) + 0 cdot (-2) = 0 + 1 + 0 = 1]Compute magnitudes:[|mathbf{n_1}| = sqrt{0^2 + (-1)^2 + 0^2} = sqrt{1} = 1][|mathbf{n_2}| = sqrt{1^2 + (-1)^2 + (-2)^2} = sqrt{1 + 1 + 4} = sqrt{6}]So,[cos theta = frac{|1|}{1 cdot sqrt{6}} = frac{1}{sqrt{6}} = frac{sqrt{6}}{6}]However, since dihedral angles are measured between 0 and 180 degrees, and the cosine can be positive or negative depending on the orientation. But since the angle between two planes is defined as the smallest angle between them, we take the absolute value. However, in our case, the dot product was positive, so the angle is acute. But depending on the orientation, sometimes the angle is considered as the obtuse one. But in this case, since the dihedral angle is formed by two planes, and the angle between their normals is acute, the dihedral angle is also acute, so the cosine is positive.But wait, let me double-check. The dihedral angle is the angle between the two planes, which can be found by the angle between their normals. However, sometimes the angle is considered as the supplement if the normals are pointing in certain directions. But in our case, since both normals are pointing in consistent directions (away from the respective planes), the angle between them is the dihedral angle.But actually, the dihedral angle is equal to the angle between the normals if the normals are pointing towards each other, otherwise, it's the supplement. Since we took the absolute value, we might have to consider the actual orientation.Wait, perhaps I should not take the absolute value. Let me think.The formula is:[cos theta = frac{mathbf{n_1} cdot mathbf{n_2}}{|mathbf{n_1}| |mathbf{n_2}|}]But depending on the orientation, the angle could be acute or obtuse. However, since we are asked for the cosine value, regardless of the angle being acute or obtuse, the cosine can be positive or negative. But in our case, the dot product is positive, so the angle is acute, so the cosine is positive.But let me verify the direction of the normals. For plane (A_1AG), the normal vector is ((0, -1, 0)), which points in the negative y-direction. For plane (EAG), the normal vector is ((1, -1, -2)). The dot product is positive, so the angle between them is acute.Therefore, the cosine of the dihedral angle is (frac{sqrt{6}}{6}).Wait, but actually, the dihedral angle is the angle between the two planes, which is equal to the angle between their normals or its supplement. Since the dot product is positive, the angle is acute, so the dihedral angle is acute, and its cosine is positive.But wait, in the problem statement, it's the dihedral angle (A_1 - AG - E). So, the edge is (AG), and the two faces are (A_1AG) and (EAG). So, the dihedral angle is the angle between these two planes along edge (AG).Given that, the angle between the normals is equal to the dihedral angle or its supplement. Since the normals are pointing in certain directions, we might have to consider the angle or its supplement.But in our case, the dot product is positive, so the angle between normals is acute, so the dihedral angle is acute, and its cosine is positive.Therefore, the cosine value is (frac{sqrt{6}}{6}).But wait, let me double-check the cross product for plane (EAG). Earlier, I computed it as ((1, -1, -2)), but let me confirm.Yes, cross product (AE times AG) was:[left( frac{s^2}{4}, -frac{s^2}{4}, -frac{s^2}{2} right )]Which simplifies to ((1, -1, -2)) when divided by (frac{s^2}{4}).So, the normal vector is correct.Therefore, the cosine of the dihedral angle is (frac{sqrt{6}}{6}).But wait, in the initial computation, the dot product was 1, and the magnitude of (mathbf{n_1}) is 1, and (mathbf{n_2}) is (sqrt{6}), so indeed, (cos theta = frac{1}{sqrt{6}} = frac{sqrt{6}}{6}).Therefore, the answer is (boxed{dfrac{sqrt{6}}{6}}).**Final Answer**1. (boxed{dfrac{1}{2}})2. Proved that (EG perp A_1C).3. (boxed{dfrac{sqrt{6}}{6}})

answer：First, recall that in an arithmetic sequence, the nth term can be expressed as:[ a_n = a_1 + (n-1)d ]where ( a_1 ) is the first term and ( d ) is the common difference.Given the equation:[ a_5 + a_9 - a_7 = 10 ]Substitute the expressions for ( a_5 ), ( a_9 ), and ( a_7 ) into the equation:[ (a_1 + 4d) + (a_1 + 8d) - (a_1 + 6d) = 10 ]Simplify the equation:[ a_1 + 4d + a_1 + 8d - a_1 - 6d = 10 ][ a_1 + 6d = 10 ][ a_7 = 10 ]The sum of the first ( n ) terms of an arithmetic sequence is given by:[ S_n = frac{n}{2} (a_1 + a_n) ]For ( S_{13} ):[ S_{13} = frac{13}{2} (a_1 + a_{13}) ]Since ( a_{13} = a_1 + 12d ), substitute this into the sum formula:[ S_{13} = frac{13}{2} (a_1 + a_1 + 12d) ][ S_{13} = frac{13}{2} (2a_1 + 12d) ][ S_{13} = 13(a_1 + 6d) ]We already found that ( a_1 + 6d = 10 ), so:[ S_{13} = 13 times 10 = 130 ]

answer：Alright, so I've got this equation here: (2^{a} + log_{2}a = 4^{b} + 2log_{4}b). And I need to figure out the relationship between (a) and (b) from the given options. Let me try to break this down step by step.First off, I notice that both sides of the equation involve exponents and logarithms with base 2 and 4. Maybe I can simplify the equation by expressing everything in terms of base 2 since 4 is a power of 2. That might make things easier to compare.Starting with the right side: (4^{b}) can be written as ((2^{2})^{b}), which simplifies to (2^{2b}). Okay, so (4^{b} = 2^{2b}). That takes care of the exponential part.Now, looking at the logarithmic term on the right side: (2log_{4}b). I remember that (log_{4}b) can be converted to base 2 using the change of base formula. The formula is (log_{c}d = frac{log_{k}d}{log_{k}c}) for any positive number (k), (c), and (d). So, applying that here, (log_{4}b = frac{log_{2}b}{log_{2}4}). Since (log_{2}4 = 2), this simplifies to (frac{log_{2}b}{2}).Therefore, (2log_{4}b = 2 times frac{log_{2}b}{2} = log_{2}b). Nice, so the right side simplifies to (2^{2b} + log_{2}b).So now, the original equation becomes:[2^{a} + log_{2}a = 2^{2b} + log_{2}b]Hmm, interesting. So both sides have an exponential term and a logarithmic term. Maybe I can compare these terms separately or find a relationship between (a) and (b) based on this.Let me think about the functions involved. The function (2^{x}) grows exponentially, while (log_{2}x) grows logarithmically. So, for larger values of (x), (2^{x}) will dominate over (log_{2}x). But since both sides have similar structures, perhaps I can analyze their growth rates.Let me consider defining a function (f(x) = 2^{x} + log_{2}x). Then, the equation becomes:[f(a) = f(2b)]Wait, is that right? Because on the right side, we have (2^{2b}), which is (2^{2b}), and (log_{2}b). So, actually, it's (f(2b)) if we think of (f(x) = 2^{x} + log_{2}x). So, (f(a) = f(2b)).Now, if (f(x)) is a strictly increasing function, then (f(a) = f(2b)) would imply that (a = 2b). But I need to check if (f(x)) is indeed strictly increasing.Let's analyze the derivative of (f(x)) to determine its monotonicity. The derivative (f'(x)) is the sum of the derivatives of (2^{x}) and (log_{2}x).The derivative of (2^{x}) with respect to (x) is (2^{x} ln 2), which is always positive for all real (x).The derivative of (log_{2}x) with respect to (x) is (frac{1}{x ln 2}), which is also positive for (x > 0).Since both terms of (f'(x)) are positive for (x > 0), (f(x)) is strictly increasing on its domain. Therefore, if (f(a) = f(2b)), it must be that (a = 2b).But wait, the original equation is (2^{a} + log_{2}a = 2^{2b} + log_{2}b). If (a = 2b), then substituting back, we get:[2^{2b} + log_{2}(2b) = 2^{2b} + log_{2}b]Simplifying (log_{2}(2b)), we have:[log_{2}2 + log_{2}b = 1 + log_{2}b]So, the equation becomes:[2^{2b} + 1 + log_{2}b = 2^{2b} + log_{2}b]Subtracting (2^{2b} + log_{2}b) from both sides, we get:[1 = 0]Wait, that doesn't make sense. So, my assumption that (a = 2b) leads to a contradiction. That means my earlier conclusion that (a = 2b) must be incorrect.Hmm, so if (f(a) = f(2b)) leads to a contradiction, perhaps (a) is not equal to (2b). Maybe (a) is less than (2b) or greater than (2b). Let's explore that.Since (f(x)) is strictly increasing, if (a < 2b), then (f(a) < f(2b)). Similarly, if (a > 2b), then (f(a) > f(2b)). But from the original equation, (f(a) = f(2b)), which suggests that (a = 2b). But that leads to a contradiction.Wait a minute, maybe I made a mistake in simplifying the equation earlier. Let me double-check.Original equation:[2^{a} + log_{2}a = 4^{b} + 2log_{4}b]Simplify (4^{b}) to (2^{2b}), and (2log_{4}b) to (log_{2}b). So, the equation becomes:[2^{a} + log_{2}a = 2^{2b} + log_{2}b]Yes, that seems correct.So, defining (f(x) = 2^{x} + log_{2}x), we have (f(a) = f(2b)). Since (f(x)) is strictly increasing, this implies (a = 2b). But when we substitute back, we get an inconsistency.This suggests that perhaps (a) is not exactly equal to (2b), but maybe slightly less or more. Let's consider that (a) might be slightly less than (2b).Suppose (a < 2b). Then, since (f(x)) is increasing, (f(a) < f(2b)). But from the equation, (f(a) = f(2b)), which would mean that (a) cannot be less than (2b).Similarly, if (a > 2b), then (f(a) > f(2b)), which again contradicts the equation (f(a) = f(2b)).This is puzzling. Maybe there's another approach.Let me consider specific values to test the relationship between (a) and (b). Suppose (b = 1). Then, the right side becomes (4^{1} + 2log_{4}1 = 4 + 2 times 0 = 4). So, the equation is (2^{a} + log_{2}a = 4).Let's solve for (a). Trying (a = 2): (2^{2} + log_{2}2 = 4 + 1 = 5), which is greater than 4. Trying (a = 1): (2^{1} + log_{2}1 = 2 + 0 = 2), which is less than 4. So, (a) is between 1 and 2 when (b = 1).Now, (2b = 2 times 1 = 2). So, (a) is less than 2b in this case.Let me try another value. Suppose (b = 2). Then, the right side is (4^{2} + 2log_{4}2 = 16 + 2 times frac{1}{2} = 16 + 1 = 17). So, the equation is (2^{a} + log_{2}a = 17).Trying (a = 4): (2^{4} + log_{2}4 = 16 + 2 = 18), which is greater than 17. Trying (a = 3): (2^{3} + log_{2}3 = 8 + approx 1.585 = approx 9.585), which is less than 17. So, (a) is between 3 and 4 when (b = 2).Here, (2b = 4), and (a) is less than 4. So again, (a < 2b).Another test: (b = 0.5). Then, the right side is (4^{0.5} + 2log_{4}0.5 = 2 + 2 times (-0.5) = 2 - 1 = 1). So, the equation is (2^{a} + log_{2}a = 1).Trying (a = 0.5): (2^{0.5} + log_{2}0.5 approx 1.414 - 1 = 0.414), which is less than 1. Trying (a = 1): (2^{1} + log_{2}1 = 2 + 0 = 2), which is greater than 1. So, (a) is between 0.5 and 1 when (b = 0.5).Here, (2b = 1), and (a) is less than 1. So again, (a < 2b).From these examples, it seems that (a) is consistently less than (2b). Maybe this is always the case.Alternatively, perhaps I can consider the function (f(x) = 2^{x} + log_{2}x) and analyze its behavior more carefully.We know that (f(x)) is strictly increasing because both (2^{x}) and (log_{2}x) are increasing functions. Therefore, if (f(a) = f(2b)), then (a = 2b). But earlier, substituting (a = 2b) led to a contradiction. So, maybe there's a mistake in assuming that (f(a) = f(2b)) directly.Wait, let's go back to the equation:[2^{a} + log_{2}a = 2^{2b} + log_{2}b]If I define (f(x) = 2^{x} + log_{2}x), then the equation is (f(a) = f(2b)). Since (f(x)) is strictly increasing, this implies (a = 2b). But when I substitute (a = 2b) back into the equation, I get:[2^{2b} + log_{2}(2b) = 2^{2b} + log_{2}b]Which simplifies to:[log_{2}(2b) = log_{2}b][log_{2}2 + log_{2}b = log_{2}b][1 + log_{2}b = log_{2}b][1 = 0]Which is impossible. Therefore, my initial assumption that (f(a) = f(2b)) implies (a = 2b) must be incorrect.This suggests that there's no solution where (a = 2b), but from the examples earlier, it seems that (a < 2b). Maybe the function (f(x)) is not strictly increasing in a way that allows (f(a) = f(2b)) unless (a < 2b).Alternatively, perhaps I need to consider the behavior of the function more carefully. Let's analyze the difference between (f(a)) and (f(2b)).Define (g(x) = f(x) - f(2b)). Then, (g(a) = 0). Since (f(x)) is strictly increasing, (g(x)) is also strictly increasing. Therefore, there can only be one solution for (g(x) = 0), which would be (x = 2b). But as we saw earlier, substituting (x = 2b) leads to a contradiction, meaning there is no solution where (a = 2b).This implies that the equation (2^{a} + log_{2}a = 2^{2b} + log_{2}b) has no solution where (a = 2b). Therefore, (a) must be either less than or greater than (2b).From the earlier examples, when (b = 1), (a) was between 1 and 2, which is less than (2b = 2). When (b = 2), (a) was between 3 and 4, which is less than (2b = 4). When (b = 0.5), (a) was between 0.5 and 1, which is less than (2b = 1).This consistent pattern suggests that (a < 2b) is the correct relationship.To further confirm, let's consider the behavior of the function (f(x) = 2^{x} + log_{2}x). Since it's strictly increasing, if (a < 2b), then (f(a) < f(2b)). But from the equation, (f(a) = f(2b)), which would mean that (a) cannot be less than (2b). However, our earlier examples showed that (a) is indeed less than (2b), leading to a contradiction.This suggests that perhaps the function (f(x)) is not strictly increasing for all (x), or there's a specific range where this relationship holds. Alternatively, maybe there's a mistake in the way I'm approaching the problem.Wait, let's consider the derivative of (f(x)) again. The derivative (f'(x) = 2^{x} ln 2 + frac{1}{x ln 2}). Both terms are positive for (x > 0), so (f(x)) is indeed strictly increasing for (x > 0). Therefore, if (f(a) = f(2b)), it must be that (a = 2b), but this leads to a contradiction.This is perplexing. Maybe the equation has no solution, but the problem states that it does, so there must be a solution where (a < 2b).Alternatively, perhaps I need to consider that (f(a) = f(2b)) implies (a = 2b) only if (f(x)) is strictly increasing and bijective, which it is. Therefore, the only solution is (a = 2b), but this leads to a contradiction, meaning there is no solution. However, the problem provides options, implying that there is a solution.This suggests that my earlier approach might be missing something. Maybe I need to consider the properties of the functions more carefully or look for another way to relate (a) and (b).Let me try to rearrange the equation:[2^{a} - 2^{2b} = log_{2}b - log_{2}a]So, the difference in the exponential terms equals the difference in the logarithmic terms.Now, if (a = 2b), then the left side becomes (2^{2b} - 2^{2b} = 0), and the right side becomes (log_{2}b - log_{2}(2b) = log_{2}b - (log_{2}2 + log_{2}b) = -1). So, (0 = -1), which is not possible.If (a < 2b), then (2^{a} < 2^{2b}), so the left side is negative. The right side is (log_{2}b - log_{2}a). If (a < 2b), then (a) is less than (2b), but we don't know how (a) compares to (b). If (a < b), then (log_{2}a < log_{2}b), making the right side positive. If (a > b), then (log_{2}a > log_{2}b), making the right side negative.So, if (a < 2b), the left side is negative. For the equation to hold, the right side must also be negative, meaning (log_{2}a > log_{2}b), which implies (a > b).Therefore, if (a < 2b) and (a > b), the equation could hold.From the earlier examples:- When (b = 1), (a) was between 1 and 2, so (a > b) and (a < 2b).- When (b = 2), (a) was between 3 and 4, so (a > b) and (a < 2b).- When (b = 0.5), (a) was between 0.5 and 1, so (a > b) and (a < 2b).This pattern suggests that (a) is always between (b) and (2b), meaning (b < a < 2b).But the options given are:A: (a > 2b)B: (a < 2b)C: (a > b^{2})D: (a < b^{2})From our analysis, (a < 2b) is always true, so option B is correct.However, let's check if (a > b^{2}) or (a < b^{2}) could also be correct.Consider (b = 1): (a) is between 1 and 2. (b^{2} = 1). So, (a > b^{2}) is true.For (b = 2): (a) is between 3 and 4. (b^{2} = 4). So, (a < b^{2}) is true.For (b = 0.5): (a) is between 0.5 and 1. (b^{2} = 0.25). So, (a > b^{2}) is true.This suggests that sometimes (a > b^{2}) and sometimes (a < b^{2}), depending on the value of (b). Therefore, we cannot definitively say that either C or D is always correct.Thus, the only consistent relationship is (a < 2b), making option B the correct answer.

answer：Okay, so I've got this problem where a large rectangle is divided into 9 smaller rectangles, kind of like a tic-tac-toe board. The areas of three of these small rectangles at the corners are given: 9, 15, and 12. I need to find the area of the fourth corner rectangle. Hmm, let me try to visualize this.First, I imagine the large rectangle divided into a 3x3 grid, making 9 smaller rectangles. The corners would be the four small rectangles at each corner of the large rectangle. So, if I label them, let's say the top-left is 9, the top-right is 15, the bottom-left is 12, and the bottom-right is the one I need to find. Or maybe the positions are different, but I think the idea is similar.I remember that in such problems, the areas often relate to each other in a proportional way because the sides of the small rectangles have to match up with each other. So, maybe the ratios of the areas can help me find the missing one.Let me denote the areas as follows: let’s say the top-left is A = 9, top-right is B = 15, bottom-left is C = 12, and bottom-right is D, which I need to find. So, A = 9, B = 15, C = 12, D = ?I think the key here is that the ratios of the areas of the small rectangles in the same row or column should be consistent. So, for example, the ratio of A to C should be the same as the ratio of B to D because they are in the same column. Alternatively, the ratio of A to B should be the same as the ratio of C to D because they are in the same row.Let me write that down:If A/C = B/D, then 9/12 = 15/D.Simplifying 9/12, that's 3/4. So, 3/4 = 15/D.Cross-multiplying, 3D = 4*15, which is 60.So, D = 60/3 = 20.Wait, that seems straightforward. But let me double-check to make sure I didn't make a mistake.Alternatively, if I consider the ratio of A to B, which is 9/15 = 3/5, and set that equal to the ratio of C to D, which is 12/D.So, 3/5 = 12/D.Cross-multiplying, 3D = 5*12 = 60.Again, D = 60/3 = 20.Okay, so both methods give me D = 20. That seems consistent.Just to make sure, maybe I can think about the sides of the rectangles. Let's say the large rectangle has length L and width W. Then, the small rectangles would have dimensions that are fractions of L and W.Suppose the top row has heights h1 and h2, and the bottom row has heights h3 and h4, but actually, since it's divided into 3 rows and 3 columns, the heights and widths would be consistent across the grid.Wait, maybe I should think in terms of variables for the sides. Let me denote the widths of the three columns as a, b, c and the heights of the three rows as d, e, f.So, the area of the top-left rectangle would be a*d = 9.The area of the top-right rectangle would be b*d = 15.The area of the bottom-left rectangle would be a*e = 12.And the area of the bottom-right rectangle would be b*e = D.So, from a*d = 9 and b*d = 15, I can find the ratio of a to b.Dividing the two equations: (a*d)/(b*d) = 9/15 => a/b = 3/5.Similarly, from a*d = 9 and a*e = 12, I can find the ratio of d to e.Dividing these: (a*d)/(a*e) = 9/12 => d/e = 3/4.Now, since the large rectangle is divided into three columns and three rows, the total width is a + b + c, and the total height is d + e + f. But I don't know anything about c and f, so maybe I don't need them.But I can express D in terms of b and e. Since D = b*e.From a/b = 3/5, we have a = (3/5)b.From d/e = 3/4, we have d = (3/4)e.Now, from a*d = 9, substituting a and d:(3/5)b * (3/4)e = 9Simplify: (9/20)be = 9Multiply both sides by 20: 9be = 180Divide by 9: be = 20But be is exactly D, so D = 20.Okay, that confirms it again. So, the area of the fourth corner rectangle is 20.I think that's solid. I approached it in two different ways: using the ratios directly and then by setting up variables for the sides. Both methods led me to the same answer, so I'm pretty confident that 20 is correct.