answer：First, I need to establish the original revenue from the tax on the commodity. Let's assume the original tax rate is ( T ) and the original consumption is ( C ). Therefore, the original revenue is ( T times C ).Next, the tax rate is decreased by 30%, which means the new tax rate is 70% of the original tax rate. Mathematically, this is represented as ( 0.70T ).At the same time, the consumption of the commodity increases by 20%, making the new consumption 120% of the original consumption. This is expressed as ( 1.20C ).The new revenue after the changes in tax rate and consumption is calculated by multiplying the new tax rate by the new consumption: ( 0.70T times 1.20C = 0.84TC ).To determine the decrease in revenue, I subtract the new revenue from the original revenue: ( TC - 0.84TC = 0.16TC ).Finally, to find the percentage decrease in revenue, I divide the decrease in revenue by the original revenue and multiply by 100%: ( frac{0.16TC}{TC} times 100% = 16% ).

answer：Alright, so I've got this problem here: If ( x - y = 16 ) and ( x + y = 8 ), what is the value of ( y )? Hmm, okay, let's see. I remember from algebra that when you have two equations with two variables, you can solve for one variable by either substitution or elimination. I think elimination might be easier here because the equations are already set up in a way that could allow me to add or subtract them to eliminate one of the variables.First, let me write down the equations again to make sure I have them right:1. ( x - y = 16 )2. ( x + y = 8 )Okay, so if I look at these two equations, I notice that both have an ( x ) and a ( y ). If I add them together, maybe the ( y ) terms will cancel out or something. Let me try that.Adding equation 1 and equation 2:( (x - y) + (x + y) = 16 + 8 )Simplifying the left side:( x - y + x + y = 2x )And the right side:( 16 + 8 = 24 )So now I have:( 2x = 24 )To solve for ( x ), I can divide both sides by 2:( x = 12 )Alright, so ( x ) is 12. Now, I need to find ( y ). I can plug this value of ( x ) back into one of the original equations to solve for ( y ). Let me choose equation 2 because it looks simpler:( x + y = 8 )Substituting ( x = 12 ):( 12 + y = 8 )To solve for ( y ), I'll subtract 12 from both sides:( y = 8 - 12 )( y = -4 )Wait, so ( y ) is -4? Let me check if this makes sense by plugging both values back into the original equations to make sure they hold true.First, equation 1:( x - y = 16 )Substituting ( x = 12 ) and ( y = -4 ):( 12 - (-4) = 12 + 4 = 16 )That works out.Now, equation 2:( x + y = 8 )Substituting ( x = 12 ) and ( y = -4 ):( 12 + (-4) = 12 - 4 = 8 )That also works out.Okay, so both equations are satisfied with ( x = 12 ) and ( y = -4 ). It looks like I did it right. I think I've got the answer.But just to make sure I didn't make any mistakes, let me try solving it another way, maybe using substitution instead of elimination.Starting again with the two equations:1. ( x - y = 16 )2. ( x + y = 8 )From equation 1, I can solve for ( x ) in terms of ( y ):( x = y + 16 )Now, substitute this expression for ( x ) into equation 2:( (y + 16) + y = 8 )Simplify:( y + 16 + y = 8 )Combine like terms:( 2y + 16 = 8 )Subtract 16 from both sides:( 2y = 8 - 16 )( 2y = -8 )Divide both sides by 2:( y = -4 )Okay, so I get the same result for ( y ) using substitution. That gives me more confidence that ( y = -4 ) is the correct answer.Just to recap, I used two different methods—elimination and substitution—and both led me to the same value for ( y ). I also checked my solution by plugging the values back into the original equations, and they both held true. So, I think I'm good.I guess the key takeaway here is that when you have two equations with two variables, you can use either elimination or substitution to solve for one variable, and then back-substitute to find the other. It's always a good idea to check your solution by plugging the values back into the original equations to ensure they satisfy both.Another thing I noticed is that in the original problem, the coefficients of ( x ) and ( y ) were set up in a way that made elimination straightforward. If the coefficients weren't as nice, I might have had to do a bit more work to make them align for elimination, but in this case, it was pretty simple.Also, it's interesting that ( y ) turned out to be negative. At first glance, I might have expected both ( x ) and ( y ) to be positive since the right-hand sides of the equations are positive, but algebra can sometimes surprise you with negative solutions. It's important not to assume the signs of the variables without working through the equations.In summary, solving systems of equations is a fundamental skill in algebra, and being comfortable with both elimination and substitution methods can be really helpful. It's also good practice to verify your solutions by plugging them back into the original equations to catch any potential mistakes.

answer：Alright, so I have this problem about a convex pentagon (ABCDE) where angles at (A), (B), and (D) are all right angles (90 degrees). I need to find the measure of angle (ADB), and it's given that a circle can be inscribed in this pentagon. Hmm, okay, inscribed circle means it's a tangential pentagon, right? So, all sides are tangent to the circle.First, let me visualize this pentagon. Since it's convex and has three right angles, it must have a sort of "L" shape or maybe something more complex. But with three right angles, it's probably not too irregular. I think drawing a rough sketch might help. Let me imagine plotting the points (A), (B), (C), (D), and (E) on a coordinate system to make it easier.Let's assume point (A) is at the origin ((0,0)). Since angle (A) is 90 degrees, sides (AB) and (AE) must be perpendicular. Let me say (AB) goes along the x-axis and (AE) goes along the y-axis. So, point (B) could be at ((a,0)) and point (E) at ((0,b)) for some positive (a) and (b).Now, angle (B) is also 90 degrees, so sides (BA) and (BC) are perpendicular. Since (BA) is along the x-axis from (A) to (B), side (BC) must go upwards from (B). Let me denote point (C) as ((a,c)), where (c > 0).Moving on, angle (D) is 90 degrees as well. So, sides (DC) and (DE) must be perpendicular. Hmm, but I don't know where point (D) is yet. Maybe I should figure out the coordinates step by step.Wait, since the pentagon is convex and has an inscribed circle, all its sides are tangent to the circle. That means the distances from the center of the circle to each side are equal, right? So, if I can find the inradius, or at least express the sides in terms of the inradius, that might help.But I'm not sure how to proceed with that yet. Maybe I should consider the properties of tangential polygons. In a tangential polygon, the sum of the lengths of every other side is equal. For a pentagon, that would mean (AB + CD + EA = BC + DE + AD). Hmm, is that correct? Wait, no, for a tangential quadrilateral, the sums of opposite sides are equal, but for a pentagon, it's more complicated.I think for a tangential polygon with an odd number of sides, there isn't a straightforward formula like in quadrilaterals. Maybe I need another approach.Let me think about the coordinates again. If I can assign coordinates to all points, I might be able to find the lengths and then use the inradius condition. Let's try that.Starting with (A(0,0)), (B(a,0)), (E(0,b)). Now, from (B(a,0)), moving to (C(a,c)). Then from (C(a,c)), moving to (D). Since angle (D) is 90 degrees, the sides (DC) and (DE) are perpendicular. So, if (C) is at ((a,c)), then (D) must be somewhere such that (DC) is either horizontal or vertical, but since (C) is at ((a,c)), and (D) is connected to (E(0,b)), which is on the y-axis, maybe (D) is at ((d,c)) or ((a,e)). Hmm, not sure.Wait, maybe I should think about the slopes. If (DC) is perpendicular to (DE), then the product of their slopes is -1. Let me denote (D) as ((d_x, d_y)). Then, the slope of (DC) is ((d_y - c)/(d_x - a)), and the slope of (DE) is ((d_y - b)/(d_x - 0)). Their product should be -1.So, (left(frac{d_y - c}{d_x - a}right) left(frac{d_y - b}{d_x}right) = -1). That's one equation.Also, since the pentagon is convex, all the points should be arranged such that the interior angles are less than 180 degrees. So, point (D) should be somewhere in the plane such that the polygon doesn't intersect itself.This is getting a bit complicated. Maybe I should use vectors or coordinate geometry to express the sides and then use the inradius condition.Alternatively, maybe I can use the fact that in a tangential polygon, the distance from the center of the incircle to each side is equal to the inradius (r). So, if I can find expressions for the distances from the center (O(h,k)) to each side, they should all equal (r).Let me denote the center as (O(h,k)). Then, the distance from (O) to side (AB) (which is along the x-axis from ((0,0)) to ((a,0))) is (k), since the distance from a point to the x-axis is its y-coordinate. Similarly, the distance from (O) to side (AE) (along the y-axis from ((0,0)) to ((0,b))) is (h), since the distance from a point to the y-axis is its x-coordinate.Since the pentagon is tangential, these distances must both equal (r). So, (h = r) and (k = r). Therefore, the center (O) is at ((r, r)).Okay, that's a useful piece of information. Now, let's find the distance from (O(r,r)) to the other sides: (BC), (CD), and (DE).First, side (BC) goes from (B(a,0)) to (C(a,c)). This is a vertical line at (x = a). The distance from (O(r,r)) to this line is (|a - r|). Since this must equal (r), we have (|a - r| = r). So, (a - r = r) or (a - r = -r). The second case would give (a = 0), which isn't possible since (a > 0). So, (a - r = r) implies (a = 2r).Similarly, let's find the distance from (O(r,r)) to side (DE). Side (DE) goes from (D(d_x, d_y)) to (E(0,b)). The equation of line (DE) can be found using the two points. Let me denote the slope as (m = (b - d_y)/(0 - d_x) = (b - d_y)/(-d_x)). So, the equation is (y - d_y = m(x - d_x)).The distance from (O(r,r)) to this line should be (r). The formula for the distance from a point ((x_0, y_0)) to the line (Ax + By + C = 0) is (|Ax_0 + By_0 + C| / sqrt{A^2 + B^2}). So, I need to write the equation of (DE) in the form (Ax + By + C = 0).Let me rearrange the equation: (y - d_y = m(x - d_x)) becomes (mx - y + ( - m d_x + d_y) = 0). So, (A = m), (B = -1), (C = -m d_x + d_y).Then, the distance from (O(r,r)) to (DE) is (|m r - r + (-m d_x + d_y)| / sqrt{m^2 + 1}) = (|m(r - d_x) + (d_y - r)| / sqrt{m^2 + 1}). This should equal (r).Hmm, this is getting quite involved. Maybe I should find expressions for (d_x) and (d_y) in terms of (r).Wait, earlier I found that (a = 2r). So, point (B) is at ((2r, 0)). Then, point (C) is at ((2r, c)). Now, let's consider side (CD). Since angle (D) is 90 degrees, sides (DC) and (DE) are perpendicular. So, the slope of (DC) times the slope of (DE) should be -1.Slope of (DC): from (C(2r, c)) to (D(d_x, d_y)) is ((d_y - c)/(d_x - 2r)).Slope of (DE): from (D(d_x, d_y)) to (E(0, b)) is ((b - d_y)/(0 - d_x) = (b - d_y)/(-d_x)).Their product should be -1:[left(frac{d_y - c}{d_x - 2r}right) left(frac{b - d_y}{-d_x}right) = -1]Simplify:[frac{(d_y - c)(b - d_y)}{-(d_x - 2r)d_x} = -1]Multiply both sides by denominator:[(d_y - c)(b - d_y) = (d_x - 2r)d_x]That's one equation.Now, let's consider the distance from (O(r,r)) to side (CD). Side (CD) goes from (C(2r, c)) to (D(d_x, d_y)). The equation of line (CD) can be written as:Slope (m_{CD} = (d_y - c)/(d_x - 2r)).Equation: (y - c = m_{CD}(x - 2r)).So, (y = m_{CD}x - 2r m_{CD} + c).Rewriting: (m_{CD}x - y - 2r m_{CD} + c = 0).Distance from (O(r,r)) to this line is:[|m_{CD} r - r - 2r m_{CD} + c| / sqrt{m_{CD}^2 + 1} = r]Simplify numerator:[| - r m_{CD} - r + c | = r sqrt{m_{CD}^2 + 1}]This seems complicated, but maybe I can express (m_{CD}) in terms of (d_x) and (d_y):(m_{CD} = (d_y - c)/(d_x - 2r)).So, substituting:[| - r left(frac{d_y - c}{d_x - 2r}right) - r + c | = r sqrt{left(frac{d_y - c}{d_x - 2r}right)^2 + 1}]This is getting really messy. Maybe there's a simpler approach.Wait, since the pentagon is tangential, all its sides are tangent to the incircle. That means each side is at a distance (r) from the center (O(r,r)). So, for side (CD), which goes from (C(2r, c)) to (D(d_x, d_y)), the distance from (O(r,r)) to this side is (r).I think I need another strategy. Maybe using the fact that in a tangential polygon, the lengths of the sides can be expressed in terms of the tangent lengths from each vertex to the points of tangency.In a tangential polygon, each side is equal to twice the radius times the tangent of half the angle at that vertex. Wait, no, that's for regular polygons. In a general tangential polygon, the lengths of the sides are related to the angles, but it's not straightforward.Alternatively, maybe I can use the Pitot theorem, which states that in a convex polygon, if the sum of the lengths of the sides is equal to twice the diameter times the number of sides, but I'm not sure.Wait, Pitot theorem actually states that for a convex polygon to be tangential, the sum of the lengths of the sides is equal to twice the diameter times the number of sides? No, that doesn't sound right.Wait, Pitot theorem says that for a convex polygon to be tangential, the sum of the lengths of the sides is equal to twice the sum of the radii of the incircle times the number of sides? Hmm, no, I think I'm confusing it.Actually, Pitot theorem states that in a convex polygon, if the sum of the distances from any interior point to the sides is constant, then the polygon is tangential. But I'm not sure if that helps here.Maybe I should consider the fact that in a tangential polygon, the lengths of the sides can be expressed as (AB = x_1 + x_2), (BC = x_2 + x_3), etc., where (x_i) are the tangent lengths from each vertex to the points of tangency.In a pentagon, this would mean:- (AB = x_1 + x_2)- (BC = x_2 + x_3)- (CD = x_3 + x_4)- (DE = x_4 + x_5)- (EA = x_5 + x_1)And since the polygon is closed, the sum of all sides equals twice the sum of all (x_i). But I'm not sure how to apply this here without knowing the side lengths.Wait, but in our case, we have right angles at (A), (B), and (D). Maybe the tangent lengths can be related to the sides.At point (A), which is a right angle, the two tangent lengths (x_1) and (x_5) should be equal because the two sides meeting at a right angle would have equal tangent lengths. Similarly, at point (B), the tangent lengths (x_2) and (x_3) should be equal, and at point (D), the tangent lengths (x_4) and (x_5') (wait, need to be careful with indices) should be equal.Wait, let me clarify. At each vertex, the two adjacent sides have tangent lengths that are equal if the angle is 90 degrees. Because in a tangential polygon with a right angle, the two tangent segments from that vertex are equal.So, at (A), (x_1 = x_5).At (B), (x_2 = x_3).At (D), which is also a right angle, the tangent lengths from (D) should be equal. Let's see, sides (CD) and (DE) meet at (D), so the tangent lengths from (D) are (x_4) and (x_5). Therefore, (x_4 = x_5).So, summarizing:- (x_1 = x_5)- (x_2 = x_3)- (x_4 = x_5)From (x_4 = x_5) and (x_1 = x_5), we get (x_1 = x_4).So, we have:- (x_1 = x_4 = x_5)- (x_2 = x_3)Let me denote (x_1 = x_4 = x_5 = p) and (x_2 = x_3 = q).Now, let's express the sides in terms of (p) and (q):- (AB = x_1 + x_2 = p + q)- (BC = x_2 + x_3 = q + q = 2q)- (CD = x_3 + x_4 = q + p)- (DE = x_4 + x_5 = p + p = 2p)- (EA = x_5 + x_1 = p + p = 2p)Wait, hold on. (EA) is from (E) to (A), which in our coordinate system is along the y-axis from ((0,b)) to ((0,0)). So, its length is (b). Similarly, (AB) is along the x-axis from ((0,0)) to ((a,0)), so its length is (a). (BC) is from ((a,0)) to ((a,c)), so its length is (c). (CD) is from ((a,c)) to (D(d_x, d_y)), so its length is (sqrt{(d_x - a)^2 + (d_y - c)^2}). (DE) is from (D(d_x, d_y)) to (E(0,b)), so its length is (sqrt{d_x^2 + (d_y - b)^2}).But from the tangent lengths, we have:- (AB = p + q = a)- (BC = 2q = c)- (CD = q + p = sqrt{(d_x - a)^2 + (d_y - c)^2})- (DE = 2p = sqrt{d_x^2 + (d_y - b)^2})- (EA = 2p = b)So, from (EA = 2p = b), we get (p = b/2).From (BC = 2q = c), we get (q = c/2).From (AB = p + q = a), substituting (p = b/2) and (q = c/2), we get (a = (b + c)/2).So, (a = (b + c)/2).Now, let's look at (CD = q + p = c/2 + b/2 = (b + c)/2 = a). So, (CD = a). But (CD) is also the distance from (C(a,c)) to (D(d_x, d_y)), which is (sqrt{(d_x - a)^2 + (d_y - c)^2} = a).So,[sqrt{(d_x - a)^2 + (d_y - c)^2} = a]Squaring both sides:[(d_x - a)^2 + (d_y - c)^2 = a^2]Similarly, (DE = 2p = b), so the distance from (D(d_x, d_y)) to (E(0,b)) is (b):[sqrt{d_x^2 + (d_y - b)^2} = b]Squaring:[d_x^2 + (d_y - b)^2 = b^2]So, now we have two equations:1. ((d_x - a)^2 + (d_y - c)^2 = a^2)2. (d_x^2 + (d_y - b)^2 = b^2)Let me expand both equations.First equation:[(d_x^2 - 2a d_x + a^2) + (d_y^2 - 2c d_y + c^2) = a^2]Simplify:[d_x^2 - 2a d_x + a^2 + d_y^2 - 2c d_y + c^2 = a^2]Subtract (a^2) from both sides:[d_x^2 - 2a d_x + d_y^2 - 2c d_y + c^2 = 0]Second equation:[d_x^2 + (d_y^2 - 2b d_y + b^2) = b^2]Simplify:[d_x^2 + d_y^2 - 2b d_y + b^2 = b^2]Subtract (b^2) from both sides:[d_x^2 + d_y^2 - 2b d_y = 0]Now, let's subtract the second equation from the first equation:[(d_x^2 - 2a d_x + d_y^2 - 2c d_y + c^2) - (d_x^2 + d_y^2 - 2b d_y) = 0 - 0]Simplify:[-2a d_x - 2c d_y + c^2 + 2b d_y = 0]Combine like terms:[-2a d_x + ( -2c + 2b ) d_y + c^2 = 0]Divide both sides by 2:[- a d_x + ( -c + b ) d_y + frac{c^2}{2} = 0]So,[- a d_x + (b - c) d_y + frac{c^2}{2} = 0]Let me write this as:[a d_x = (b - c) d_y + frac{c^2}{2}]So,[d_x = frac{(b - c) d_y + frac{c^2}{2}}{a}]But earlier, we found that (a = frac{b + c}{2}). So, substitute (a = frac{b + c}{2}):[d_x = frac{(b - c) d_y + frac{c^2}{2}}{frac{b + c}{2}} = frac{2(b - c) d_y + c^2}{b + c}]So,[d_x = frac{2(b - c) d_y + c^2}{b + c}]Now, let's substitute this expression for (d_x) into the second equation:[d_x^2 + d_y^2 - 2b d_y = 0]Substitute (d_x):[left( frac{2(b - c) d_y + c^2}{b + c} right)^2 + d_y^2 - 2b d_y = 0]This looks complicated, but let's try to expand it.Let me denote (N = 2(b - c) d_y + c^2) and (D = b + c), so (d_x = N/D).Then,[left( frac{N}{D} right)^2 + d_y^2 - 2b d_y = 0]Multiply through by (D^2) to eliminate denominators:[N^2 + D^2 d_y^2 - 2b D^2 d_y = 0]Now, expand (N^2):[[2(b - c) d_y + c^2]^2 = 4(b - c)^2 d_y^2 + 4(b - c)c^2 d_y + c^4]So, the equation becomes:[4(b - c)^2 d_y^2 + 4(b - c)c^2 d_y + c^4 + (b + c)^2 d_y^2 - 2b(b + c)^2 d_y = 0]Let me collect like terms.First, the (d_y^2) terms:[4(b - c)^2 d_y^2 + (b + c)^2 d_y^2 = [4(b^2 - 2bc + c^2) + (b^2 + 2bc + c^2)] d_y^2 = [4b^2 - 8bc + 4c^2 + b^2 + 2bc + c^2] d_y^2 = (5b^2 - 6bc + 5c^2) d_y^2]Next, the (d_y) terms:[4(b - c)c^2 d_y - 2b(b + c)^2 d_y = [4c^2(b - c) - 2b(b^2 + 2bc + c^2)] d_y]Let me compute each part:(4c^2(b - c) = 4b c^2 - 4c^3)(2b(b^2 + 2bc + c^2) = 2b^3 + 4b^2 c + 2b c^2)So, subtracting:[4b c^2 - 4c^3 - 2b^3 - 4b^2 c - 2b c^2 = (-2b^3 - 4b^2 c + 2b c^2 - 4c^3)]So, the (d_y) term is:[(-2b^3 - 4b^2 c + 2b c^2 - 4c^3) d_y]Finally, the constant term is (c^4).Putting it all together:[(5b^2 - 6bc + 5c^2) d_y^2 + (-2b^3 - 4b^2 c + 2b c^2 - 4c^3) d_y + c^4 = 0]This is a quadratic equation in (d_y). Let me write it as:[A d_y^2 + B d_y + C = 0]Where:- (A = 5b^2 - 6bc + 5c^2)- (B = -2b^3 - 4b^2 c + 2b c^2 - 4c^3)- (C = c^4)This quadratic equation can be solved for (d_y):[d_y = frac{-B pm sqrt{B^2 - 4AC}}{2A}]But this seems extremely complicated. Maybe there's a simpler relationship or perhaps some symmetry I can exploit.Wait, earlier I found that (a = (b + c)/2). Maybe I can assume specific values for (b) and (c) to simplify the problem. Since the problem doesn't give specific lengths, the angle (ADB) should be the same regardless of the actual lengths, as long as the conditions are satisfied.Let me assume (b = c). Then, (a = (b + b)/2 = b). So, (a = b = c). Let's see if this assumption holds.If (b = c), then (a = b). So, point (B) is at ((b, 0)), point (C) is at ((b, b)). Then, let's see what happens to the equations.From earlier, (d_x = frac{2(b - c) d_y + c^2}{b + c}). If (b = c), this becomes:[d_x = frac{2(0) d_y + b^2}{2b} = frac{b^2}{2b} = frac{b}{2}]So, (d_x = b/2).Now, substitute (d_x = b/2) into the second equation:[d_x^2 + (d_y - b)^2 = b^2]Which becomes:[left(frac{b}{2}right)^2 + (d_y - b)^2 = b^2]Simplify:[frac{b^2}{4} + (d_y - b)^2 = b^2]Subtract (b^2/4):[(d_y - b)^2 = frac{3b^2}{4}]Take square roots:[d_y - b = pm frac{sqrt{3}b}{2}]So,[d_y = b pm frac{sqrt{3}b}{2}]Since the pentagon is convex, point (D) must lie above the line (BC), which is at (y = b). So, (d_y > b). Therefore,[d_y = b + frac{sqrt{3}b}{2}]So, (d_y = b left(1 + frac{sqrt{3}}{2}right)).Therefore, point (D) is at ((b/2, b(1 + sqrt{3}/2))).Now, let's find angle (ADB). Points (A(0,0)), (D(b/2, b(1 + sqrt{3}/2))), and (B(b,0)).We can find vectors (DA) and (DB) and then find the angle between them.Vector (DA = A - D = (0 - b/2, 0 - b(1 + sqrt{3}/2)) = (-b/2, -b(1 + sqrt{3}/2))).Vector (DB = B - D = (b - b/2, 0 - b(1 + sqrt{3}/2)) = (b/2, -b(1 + sqrt{3}/2))).The angle between vectors (DA) and (DB) can be found using the dot product:[cos theta = frac{DA cdot DB}{|DA| |DB|}]Compute (DA cdot DB):[(-b/2)(b/2) + (-b(1 + sqrt{3}/2))(-b(1 + sqrt{3}/2)) = (-b^2/4) + b^2(1 + sqrt{3}/2)^2]Compute (|DA|):[sqrt{(-b/2)^2 + (-b(1 + sqrt{3}/2))^2} = sqrt{b^2/4 + b^2(1 + sqrt{3}/2)^2}]Similarly, (|DB| = |DA|) because the vectors are symmetric.Let me compute (DA cdot DB):First term: (-b^2/4)Second term: (b^2(1 + sqrt{3}/2)^2)Compute ((1 + sqrt{3}/2)^2 = 1 + sqrt{3} + 3/4 = (1 + 3/4) + sqrt{3} = 7/4 + sqrt{3})So, second term: (b^2(7/4 + sqrt{3}))Thus, (DA cdot DB = -b^2/4 + b^2(7/4 + sqrt{3}) = b^2(-1/4 + 7/4 + sqrt{3}) = b^2(6/4 + sqrt{3}) = b^2(3/2 + sqrt{3}))Now, compute (|DA|):[sqrt{b^2/4 + b^2(7/4 + sqrt{3})} = sqrt{b^2/4 + 7b^2/4 + b^2 sqrt{3}} = sqrt{8b^2/4 + b^2 sqrt{3}} = sqrt{2b^2 + b^2 sqrt{3}} = b sqrt{2 + sqrt{3}}]So, (|DA| = |DB| = b sqrt{2 + sqrt{3}})Therefore,[cos theta = frac{b^2(3/2 + sqrt{3})}{(b sqrt{2 + sqrt{3}})^2} = frac{(3/2 + sqrt{3})}{2 + sqrt{3}}]Simplify the denominator:Multiply numerator and denominator by (2 - sqrt{3}):[frac{(3/2 + sqrt{3})(2 - sqrt{3})}{(2 + sqrt{3})(2 - sqrt{3})} = frac{(3/2)(2) - (3/2)sqrt{3} + 2sqrt{3} - (sqrt{3})^2}{4 - 3} = frac{3 - (3/2)sqrt{3} + 2sqrt{3} - 3}{1}]Simplify numerator:[3 - 3 + (- (3/2)sqrt{3} + 2sqrt{3}) = (0) + ( ( - 3/2 + 2 ) sqrt{3} ) = (1/2) sqrt{3}]So,[cos theta = frac{sqrt{3}/2}{1} = sqrt{3}/2]Therefore, (theta = 30^circ).Wait, but that's the angle between vectors (DA) and (DB). However, angle (ADB) is the angle at point (D), which is the angle between vectors (DA) and (DB). So, (theta = 30^circ).But wait, in my assumption, I set (b = c), which might not hold in the general case. However, since the problem didn't specify any particular lengths, and the angle came out to 30 degrees, which is a common angle, maybe that's the answer.But wait, earlier I thought the answer might be 45 degrees. Hmm, conflicting results.Wait, let me double-check my calculations. When I assumed (b = c), I found that angle (ADB) is 30 degrees. But maybe this is specific to that assumption. Let me see if there's another way.Alternatively, perhaps the angle is 45 degrees. Let me think about the properties of the pentagon. Since it's tangential and has three right angles, maybe the diagonals create isosceles right triangles, leading to 45-degree angles.Wait, another approach: in a tangential polygon with an incircle, the center of the circle is equidistant from all sides. So, in our case, the center (O(r,r)) is at equal distances from all sides.Given that, maybe the triangle (ADB) is such that (O) is the circumcenter, making triangle (ADB) a right-angled triangle with (AO = BO = DO), implying that angle (ADB) is 45 degrees.Wait, that seems plausible. If (O) is the circumcenter, then triangle (ADB) is inscribed in a circle with center (O), and since (AO = BO = DO), the triangle is equilateral? No, not necessarily, but the angles subtended by the chords would be related.Wait, actually, if (AO = BO = DO), then points (A), (B), and (D) lie on a circle centered at (O). Therefore, angle (ADB) is half the measure of the central angle (AOB).But what's angle (AOB)? Points (A(0,0)), (O(r,r)), (B(a,0)). The angle at (O) between (OA) and (OB).Vector (OA) is from (O(r,r)) to (A(0,0)): ((-r, -r)).Vector (OB) is from (O(r,r)) to (B(a,0)): ((a - r, -r)).The angle between these vectors can be found using the dot product:[cos phi = frac{(-r)(a - r) + (-r)(-r)}{sqrt{r^2 + r^2} sqrt{(a - r)^2 + r^2}} = frac{-r(a - r) + r^2}{sqrt{2r^2} sqrt{(a - r)^2 + r^2}}]Simplify numerator:[-r(a - r) + r^2 = -ar + r^2 + r^2 = -ar + 2r^2]Denominator:[sqrt{2r^2} sqrt{(a - r)^2 + r^2} = r sqrt{2} sqrt{(a - r)^2 + r^2}]So,[cos phi = frac{-ar + 2r^2}{r sqrt{2} sqrt{(a - r)^2 + r^2}} = frac{-a + 2r}{sqrt{2} sqrt{(a - r)^2 + r^2}}]But earlier, we found that (a = 2r). So, substitute (a = 2r):[cos phi = frac{-2r + 2r}{sqrt{2} sqrt{(2r - r)^2 + r^2}} = frac{0}{sqrt{2} sqrt{r^2 + r^2}} = 0]Therefore, angle (phi = 90^circ).So, the central angle (AOB) is 90 degrees. Therefore, the inscribed angle (ADB) subtended by the same arc is half of that, which is (45^circ).Therefore, angle (ADB = 45^circ).This contradicts my earlier result when I assumed (b = c). But since in the general case, the central angle is 90 degrees, leading to an inscribed angle of 45 degrees, I think this is the correct answer.So, despite the specific case giving 30 degrees, the general case with (a = 2r) leads to angle (ADB = 45^circ).**Final Answer**The measure of angle (ADB) is boxed{45^circ}.

answer：First, I need to determine the value of ( f(2) ). Since ( 2 ) is greater than or equal to ( 1 ), I will use the second part of the piecewise function, which is ( -2x + 3 ).Calculating ( f(2) ):[f(2) = -2(2) + 3 = -4 + 3 = -1]Next, I need to find ( f(f(2)) ), which is ( f(-1) ). Since ( -1 ) is less than ( 1 ), I will use the first part of the piecewise function, which is ( x^2 + 1 ).Calculating ( f(-1) ):[f(-1) = (-1)^2 + 1 = 1 + 1 = 2]Therefore, the value of ( f(f(2)) ) is ( 2 ).

answer：Okay, so I have this question here: "Right angles are equal" and "Equal angles are right angles" are (　　). The options are A: Inverse propositions to each other, B: Inverse theorems, C: Axioms, D: False propositions. I need to figure out which one is correct.First, let me understand what each of these terms means. An inverse proposition, if I remember correctly, is when you switch the hypothesis and conclusion of a conditional statement. So, if the original statement is "If P, then Q," the inverse would be "If not P, then not Q." Wait, no, that's actually the contrapositive. Hmm, maybe I'm mixing things up. Let me think again.Oh, right, the inverse is when you negate both the hypothesis and the conclusion. So, for "If P, then Q," the inverse would be "If not P, then not Q." But in this case, the two statements given are "Right angles are equal" and "Equal angles are right angles." These don't seem to be negations; they seem more like converse statements.Wait, the converse is when you switch the hypothesis and conclusion without negating them. So, if the original is "If P, then Q," the converse is "If Q, then P." That seems to fit here. "Right angles are equal" can be phrased as "If an angle is a right angle, then it is equal to other right angles." The converse would be "If angles are equal, then they are right angles." So, these two statements are converse of each other, not inverse.But the options don't have converse; they have inverse propositions. Hmm, maybe I need to revisit the definitions. Let me check:- **Inverse Propositions**: These are formed by negating both the hypothesis and conclusion of the original statement. So, if the original is "P implies Q," the inverse is "not P implies not Q."- **Converse Propositions**: These are formed by swapping the hypothesis and conclusion. So, "Q implies P."In the given problem, the first statement is "Right angles are equal." Let's break that down. It's like saying "If an angle is a right angle, then it is equal to other right angles." The second statement is "Equal angles are right angles," which is "If angles are equal, then they are right angles." So, the first statement is "P implies Q," where P is "an angle is a right angle" and Q is "it is equal to other right angles." The second statement is "Q implies P," which is the converse, not the inverse. Therefore, they are converse propositions, not inverse.But the options don't have converse; they have inverse. So, maybe I'm misunderstanding something. Let me think differently.Perhaps the question is considering "Right angles are equal" as a statement, and "Equal angles are right angles" as another. Are these inverses? If we consider the first as "All right angles are equal," and the second as "All equal angles are right angles," then they are not inverses but converses.Wait, maybe in some contexts, people refer to converse as inverse? No, that doesn't seem right. Inverse should involve negation. So, if the original is "Right angles are equal," the inverse would be "Non-right angles are unequal." But that's not what's given here.So, perhaps the question is incorrect in its phrasing, or I'm misinterpreting it. Let me try to see if these statements are actually inverse theorems or something else.Inverse theorems would mean that each theorem is the inverse of the other, but again, that involves negation. Since neither statement involves negation, they are not inverse theorems.Axioms are statements accepted without proof, but these are specific about right angles and equality, which are theorems, not axioms.False propositions: Are these statements false? Well, "Right angles are equal" is true because all right angles measure 90 degrees, so they are equal. "Equal angles are right angles" is not necessarily true because angles can be equal and not right angles, like two 45-degree angles. So, the second statement is false.Therefore, the second statement is a false proposition, but the first is true. So, they are not both axioms or inverse theorems. Since one is true and the other is false, they can't be inverse propositions in the sense of logical inverses, which would both involve negation.Wait, but the question is asking about the relationship between the two statements. Since one is the converse of the other, and one is true while the other is false, they are not inverse propositions. So, maybe the correct answer is D: False propositions, but only the second one is false, not both.Hmm, this is confusing. Let me try to approach it differently. Maybe the question is referring to the logical structure. If "Right angles are equal" is a proposition, then "Equal angles are right angles" is its converse. In logic, the converse is not the same as the inverse. The inverse would be "If not P, then not Q," which is different.So, if the question is considering whether they are inverse propositions, that would be incorrect because they are actually converse propositions. Since the options don't include converse, but include inverse propositions, inverse theorems, axioms, or false propositions, I need to see which fits.Given that one is true and the other is false, they are not axioms because axioms are always true. They are not inverse theorems because inverse theorems would require negation. They are not both false propositions because the first is true. Therefore, the correct answer might be that they are not inverse propositions, but since that's not an option, perhaps the closest is A: Inverse propositions to each other, but that's not accurate.Alternatively, since the second statement is false, maybe the answer is D: False propositions, but only the second is false. The question says "are (　　)," implying both together. Since one is true and one is false, they are not both false. So, D is not correct.I'm getting stuck here. Maybe I should look up the definitions again.- **Inverse Proposition**: Formed by negating both the hypothesis and conclusion of the original statement.- **Converse Proposition**: Formed by swapping the hypothesis and conclusion.- **Inverse Theorem**: A theorem that is the inverse of another theorem.- **Axiom**: A statement accepted as true without proof.- **False Proposition**: A statement that is not true.Given that, "Right angles are equal" is a theorem, not an axiom, because it can be proven. "Equal angles are right angles" is not necessarily true, so it's a false proposition.Therefore, the two statements are not inverse propositions or theorems. One is a theorem, the other is a false proposition. So, they are not related as inverse propositions or theorems. They are not axioms because they are not universally accepted without proof. One is a theorem, the other is false.But the question is asking what they are together. Since they are not inverse propositions, not inverse theorems, not axioms, and not both false propositions, perhaps the answer is D: False propositions, but only the second is false. However, the first is true, so they can't both be false.Wait, maybe the question is in Chinese, and the translation is slightly off. The original question is in Chinese, and the user provided the translation. Maybe "Right angles are equal" and "Equal angles are right angles" are being considered as inverse propositions because they swap the terms, even though one is true and the other is false.In logic, inverse propositions involve negation, but here it's just swapping. So, they are converse, not inverse. Therefore, the answer should be that they are converse propositions, but since that's not an option, perhaps the closest is A: Inverse propositions to each other, but that's not accurate.Alternatively, since one is true and the other is false, they are not equivalent, so they can't be inverse theorems. They are not axioms. Therefore, the correct answer might be D: False propositions, but only the second is false.But the question is about their relationship, not their truth value. So, maybe the answer is A: Inverse propositions to each other, even though technically they are converse. Perhaps in the context of the question, they consider swapping as inverse.I'm a bit confused, but given the options, I think the intended answer is A: Inverse propositions to each other, even though strictly speaking, they are converse. Maybe in the context of the question, inverse is used to mean converse.Alternatively, since the second statement is false, maybe the answer is D: False propositions, but only the second is false, so that's not correct.I think I'll go with A: Inverse propositions to each other, even though technically they are converse, because the question seems to consider swapping as inverse.

answer：Okay, so I have this integral to solve: the integral of cos(2x) divided by (cos²x times sin²x) dx. Hmm, that looks a bit complicated, but maybe I can simplify it somehow. Let me think about trigonometric identities that might help here.First, I remember that cos(2x) can be expressed in terms of cos²x and sin²x. There are a few identities for cos(2x). One of them is cos(2x) = cos²x - sin²x. That might be useful here because I see both cos²x and sin²x in the denominator. Let me try substituting that in.So, replacing cos(2x) with cos²x - sin²x, the integral becomes:∫ (cos²x - sin²x) / (cos²x sin²x) dxNow, I can split this fraction into two separate terms:∫ [cos²x / (cos²x sin²x) - sin²x / (cos²x sin²x)] dxSimplifying each term individually, the first term is cos²x divided by cos²x sin²x. The cos²x cancels out, leaving 1/sin²x. Similarly, the second term is sin²x divided by cos²x sin²x, and the sin²x cancels out, leaving 1/cos²x. So now the integral becomes:∫ [1/sin²x - 1/cos²x] dxI know that 1/sin²x is the same as csc²x, and 1/cos²x is the same as sec²x. So, rewriting the integral:∫ csc²x dx - ∫ sec²x dxNow, I need to remember the integrals of csc²x and sec²x. I think the integral of csc²x is -cotx + C, and the integral of sec²x is tanx + C. Let me double-check that. Yes, the derivative of cotx is -csc²x, so integrating csc²x gives -cotx, and the derivative of tanx is sec²x, so integrating sec²x gives tanx.So, putting it all together:∫ csc²x dx - ∫ sec²x dx = -cotx - tanx + CWait, is that right? Let me make sure I didn't mix up any signs. The integral of csc²x is indeed -cotx, and the integral of sec²x is tanx. So subtracting tanx gives us -tanx. So combining them, it should be -cotx - tanx + C.Hmm, let me see if there's another way to approach this integral to confirm my answer. Maybe using substitution or another identity. Let's try expressing everything in terms of sinx and cosx.Starting again, the original integral is:∫ cos(2x) / (cos²x sin²x) dxI can write cos(2x) as 1 - 2sin²x or 2cos²x - 1, but I'm not sure if that helps more than the previous approach. Maybe another substitution? Let me think.Alternatively, since the denominator is cos²x sin²x, which is (1/4)sin²(2x) because sin(2x) = 2sinx cosx, so sin²(2x) = 4sin²x cos²x. Therefore, cos²x sin²x = (1/4)sin²(2x). So, substituting that in, the integral becomes:∫ cos(2x) / [(1/4)sin²(2x)] dx = 4 ∫ cos(2x) / sin²(2x) dxNow, let me make a substitution. Let u = sin(2x), then du/dx = 2cos(2x), so (1/2)du = cos(2x) dx. Substituting, the integral becomes:4 ∫ [cos(2x) dx] / sin²(2x) = 4 ∫ (1/2 du) / u² = 2 ∫ u⁻² duIntegrating u⁻² gives -u⁻¹ + C, so:2 * (-1/u) + C = -2/u + C = -2/sin(2x) + CBut wait, earlier I got -cotx - tanx + C. Are these two expressions equivalent? Let me check.Expressing -2/sin(2x) in terms of cotx and tanx. Since sin(2x) = 2sinx cosx, so -2/sin(2x) = -2/(2sinx cosx) = -1/(sinx cosx). On the other hand, -cotx - tanx = -(cosx/sinx) - (sinx/cosx) = -(cos²x + sin²x)/(sinx cosx) = -1/(sinx cosx) because cos²x + sin²x = 1. So yes, both expressions are equal. Therefore, both methods give the same result.So, the integral is equal to -cotx - tanx + C, or equivalently, -2/sin(2x) + C. But since the problem didn't specify a particular form, either should be acceptable. However, the first form in terms of cotx and tanx might be more straightforward.Let me just recap the steps to ensure I didn't make any mistakes:1. Recognized that cos(2x) can be expressed as cos²x - sin²x.2. Split the integral into two simpler integrals.3. Integrated each term separately, recalling the integrals of csc²x and sec²x.4. Verified the result using a substitution method, confirming that both approaches lead to equivalent answers.Therefore, I'm confident that the integral evaluates to -cotx - tanx + C.