💡Okay, so I have this problem about a quadrilateral ABCD. Points K and L are the trisection points of sides AB and AD, closer to A. Similarly, points N and M are the trisection points of sides CB and CD, closer to C. I need to find the ratio of the areas of quadrilaterals ABCD and KLMN.Hmm, let me try to visualize this. So, ABCD is a quadrilateral, which could be any four-sided figure, right? It might be convex or concave, but I guess the problem doesn't specify, so I might need to consider both cases.First, let me recall what trisection points mean. If K and L are trisection points closer to A, that means they divide AB and AD into three equal parts, with AK = KB/2 and AL = LD/2, right? Similarly, N and M divide CB and CD into three equal parts, closer to C, so CN = NB/2 and CM = MD/2.Wait, actually, if they are trisection points closer to A, then AK = (1/3)AB and AL = (1/3)AD. Similarly, CN = (1/3)CB and CM = (1/3)CD. Yeah, that makes sense because trisection points divide the side into three equal segments.So, points K and L are each 1/3 of the way from A to B and A to D, respectively. Similarly, points N and M are each 1/3 of the way from C to B and C to D, respectively.Now, quadrilateral KLMN is formed by connecting these points. I need to find the ratio of the areas of ABCD and KLMN.I think the best way to approach this is to use coordinate geometry. Maybe I can assign coordinates to the vertices of ABCD and then find the coordinates of K, L, M, and N. Then, I can calculate the areas of both quadrilaterals.Let me assign coordinates to ABCD. Let's assume point A is at (0, 0). Let me denote the coordinates as follows:- A: (0, 0)- B: (b, 0)- D: (0, d)- C: (c, d)Wait, is that a good assumption? If I place A at the origin, and since AB is a side, I can place B at (b, 0). Similarly, since AD is another side, I can place D at (0, d). Then, point C would be at (c, d) to complete the quadrilateral.But actually, in a general quadrilateral, the coordinates might not be so straightforward. Maybe I should use vectors or affine coordinates.Alternatively, I can use the concept of similar triangles and area ratios without assigning specific coordinates.Let me think about the areas. Since K and L are trisection points closer to A, the segments AK and AL are each 1/3 of AB and AD, respectively. Similarly, CN and CM are 1/3 of CB and CD, respectively.So, perhaps I can divide the quadrilateral ABCD into smaller regions whose areas I can relate to the area of KLMN.Wait, maybe using the concept of similar triangles and area ratios. If I can express KLMN in terms of ABCD by subtracting certain areas, then I can find the ratio.Alternatively, I can use the shoelace formula once I have coordinates for all points.Let me try assigning coordinates again, but more carefully.Let me place point A at (0, 0). Let me denote:- Point A: (0, 0)- Point B: (3, 0) [choosing 3 for simplicity since we have trisection points]- Point D: (0, 3) [again, choosing 3 for simplicity]- Point C: (3, 3) [completing the square, but actually, ABCD doesn't have to be a square]Wait, but if I make ABCD a square, then KLMN would be a smaller square inside, and the ratio would be easier to compute. But the problem is about a general quadrilateral, so maybe I shouldn't assume it's a square.Alternatively, I can assign coordinates such that ABCD is a parallelogram, which might simplify the calculations.Let me try that. Let me assume ABCD is a parallelogram with A at (0, 0), B at (3, 0), D at (0, 3), and C at (3, 3). So, it's a square in this case.Now, points K and L are the trisection points closer to A. So, K is 1/3 from A to B, which would be at (1, 0). Similarly, L is 1/3 from A to D, which would be at (0, 1).Points N and M are the trisection points closer to C. So, N is 1/3 from C to B. Since C is at (3, 3) and B is at (3, 0), moving 1/3 from C towards B would be at (3, 2). Similarly, M is 1/3 from C to D. Since C is at (3, 3) and D is at (0, 3), moving 1/3 from C towards D would be at (2, 3).So, quadrilateral KLMN has vertices at K(1, 0), L(0, 1), M(2, 3), and N(3, 2).Now, let's compute the area of ABCD. Since it's a square with side length 3, the area is 9.Now, let's compute the area of KLMN using the shoelace formula.The coordinates of KLMN are:K: (1, 0)L: (0, 1)M: (2, 3)N: (3, 2)Let me list them in order:(1, 0), (0, 1), (2, 3), (3, 2), (1, 0)Shoelace formula:Area = 1/2 |sum over i (x_i y_{i+1} - x_{i+1} y_i)|Compute each term:1. (1)(1) - (0)(0) = 1 - 0 = 12. (0)(3) - (2)(1) = 0 - 2 = -23. (2)(2) - (3)(3) = 4 - 9 = -54. (3)(0) - (1)(2) = 0 - 2 = -2Sum these up: 1 - 2 - 5 - 2 = -8Take absolute value and multiply by 1/2: 1/2 * 8 = 4So, the area of KLMN is 4.Therefore, the ratio of the areas ABCD to KLMN is 9:4, or 9/4.Wait, but the problem asks for the ratio of ABCD to KLMN, so 9/4.But hold on, in this specific case, ABCD is a square. But the problem is about a general quadrilateral. So, does this ratio hold for any quadrilateral?I think it does because the ratio is based on the trisection points and the way the areas are divided, which should be consistent regardless of the specific shape of the quadrilateral.Alternatively, maybe I should test it with a different quadrilateral to confirm.Let me try with a different quadrilateral, say, a trapezoid.Let me assign coordinates as follows:- A: (0, 0)- B: (3, 0)- D: (0, 3)- C: (1, 3)So, ABCD is a trapezoid with bases AB = 3 and CD = 1, and height 3.Compute area of ABCD: The area of a trapezoid is (base1 + base2)/2 * height = (3 + 1)/2 * 3 = 4/2 * 3 = 6.Now, find points K, L, M, N.K is 1/3 from A to B: (1, 0)L is 1/3 from A to D: (0, 1)N is 1/3 from C to B: C is (1,3), B is (3,0). The vector from C to B is (2, -3). 1/3 of that is (2/3, -1). So, N is at (1 + 2/3, 3 - 1) = (5/3, 2)M is 1/3 from C to D: C is (1,3), D is (0,3). The vector from C to D is (-1, 0). 1/3 of that is (-1/3, 0). So, M is at (1 - 1/3, 3 + 0) = (2/3, 3)So, quadrilateral KLMN has vertices at K(1,0), L(0,1), M(2/3,3), N(5/3,2)Compute area using shoelace formula.List the coordinates in order:(1,0), (0,1), (2/3,3), (5/3,2), (1,0)Compute each term:1. (1)(1) - (0)(0) = 1 - 0 = 12. (0)(3) - (2/3)(1) = 0 - 2/3 = -2/33. (2/3)(2) - (5/3)(3) = 4/3 - 5 = 4/3 - 15/3 = -11/34. (5/3)(0) - (1)(2) = 0 - 2 = -2Sum these up: 1 - 2/3 - 11/3 - 2Convert to thirds:1 = 3/3So, 3/3 - 2/3 - 11/3 - 6/3 = (3 - 2 - 11 - 6)/3 = (-16)/3Take absolute value and multiply by 1/2: 1/2 * 16/3 = 8/3So, area of KLMN is 8/3.Area of ABCD is 6.Ratio ABCD to KLMN: 6 / (8/3) = 6 * 3/8 = 18/8 = 9/4.Same ratio as before. So, it seems consistent.Therefore, regardless of the quadrilateral, the ratio is 9/4.Wait, but in the first case, ABCD was a square with area 9, KLMN was 4, ratio 9/4. In the second case, ABCD was a trapezoid with area 6, KLMN was 8/3, ratio 9/4.So, seems like the ratio is consistent.Therefore, the ratio of the areas of quadrilaterals ABCD to KLMN is 9/4.But wait, the problem says "quadrilaterals ABCD and KLMN". So, it's ABCD : KLMN, which is 9:4, or 9/4.So, the answer is 9/4.But let me think again. Maybe I made a mistake in assuming the coordinates. Let me try another quadrilateral.Let me take a convex quadrilateral that's not a trapezoid or square.Let me assign coordinates:- A: (0,0)- B: (3,0)- D: (0,3)- C: (2,2)So, ABCD has vertices at (0,0), (3,0), (2,2), (0,3)Compute area of ABCD using shoelace formula.List the coordinates:(0,0), (3,0), (2,2), (0,3), (0,0)Compute terms:1. (0)(0) - (3)(0) = 0 - 0 = 02. (3)(2) - (2)(0) = 6 - 0 = 63. (2)(3) - (0)(2) = 6 - 0 = 64. (0)(0) - (0)(3) = 0 - 0 = 0Sum: 0 + 6 + 6 + 0 = 12Area = 1/2 |12| = 6Now, find points K, L, M, N.K is 1/3 from A to B: (1,0)L is 1/3 from A to D: (0,1)N is 1/3 from C to B: C is (2,2), B is (3,0). Vector CB is (1, -2). 1/3 of that is (1/3, -2/3). So, N is at (2 + 1/3, 2 - 2/3) = (7/3, 4/3)M is 1/3 from C to D: C is (2,2), D is (0,3). Vector CD is (-2,1). 1/3 of that is (-2/3, 1/3). So, M is at (2 - 2/3, 2 + 1/3) = (4/3, 7/3)So, quadrilateral KLMN has vertices at K(1,0), L(0,1), M(4/3,7/3), N(7/3,4/3)Compute area using shoelace formula.List the coordinates:(1,0), (0,1), (4/3,7/3), (7/3,4/3), (1,0)Compute terms:1. (1)(1) - (0)(0) = 1 - 0 = 12. (0)(7/3) - (4/3)(1) = 0 - 4/3 = -4/33. (4/3)(4/3) - (7/3)(7/3) = 16/9 - 49/9 = -33/9 = -11/34. (7/3)(0) - (1)(4/3) = 0 - 4/3 = -4/3Sum these up: 1 - 4/3 - 11/3 - 4/3Convert 1 to 3/3:3/3 - 4/3 - 11/3 - 4/3 = (3 - 4 - 11 - 4)/3 = (-16)/3Take absolute value and multiply by 1/2: 1/2 * 16/3 = 8/3So, area of KLMN is 8/3.Area of ABCD is 6.Ratio ABCD to KLMN: 6 / (8/3) = 6 * 3/8 = 18/8 = 9/4.Again, the same ratio.Therefore, it seems that regardless of the specific quadrilateral, the ratio is consistently 9/4.So, I think the answer is 9/4.**Final Answer**The ratio of the areas of quadrilaterals (ABCD) and (KLMN) is boxed{dfrac{9}{4}}.

💡Alright, so I have this problem here with a circle and a line, and I need to figure out which of the statements A, B, C, or D are true. Let me take it step by step.First, the circle is given by the equation ( x^2 + y^2 - 2x - 8 = 0 ). I remember that the general equation of a circle is ( x^2 + y^2 + Dx + Ey + F = 0 ), and its center is at ( (-D/2, -E/2) ) with radius ( sqrt{(D/2)^2 + (E/2)^2 - F} ). So, for this circle, D is -2 and E is 0. That means the center should be at ( (-(-2)/2, -0/2) = (1, 0) ). Okay, so statement A says the center is (1, 0), which seems correct. I'll tentatively say A is true.Next, statement B: The point (-1, 1) lies on line l. The line l is given by ( y = k(x + 1) + 1 ). Let me plug in x = -1 and y = 1 into this equation. So, 1 = k(-1 + 1) + 1, which simplifies to 1 = k(0) + 1, so 1 = 1. That's always true, regardless of k. So, no matter what k is, the point (-1, 1) is on line l. So, B is definitely true.Moving on to statement C: Line l intersects circle C. Hmm, to check if a line intersects a circle, one way is to solve the system of equations and see if there are real solutions. Alternatively, I can find the distance from the center of the circle to the line and see if it's less than or equal to the radius.The center of the circle is (1, 0), and the radius is... let me calculate that. From the equation, radius squared is ( (D/2)^2 + (E/2)^2 - F ). So, ( ( -2/2 )^2 + (0/2)^2 - (-8) = (1)^2 + 0 + 8 = 1 + 8 = 9 ). So, the radius is 3.Now, the distance from the center (1, 0) to the line l: ( y = k(x + 1) + 1 ). Let me rewrite this in standard form. Subtract y: ( kx - y + k + 1 = 0 ). So, the distance from (1, 0) to this line is ( |k*1 - 1*0 + k + 1| / sqrt{k^2 + (-1)^2} ) which simplifies to ( |2k + 1| / sqrt{k^2 + 1} ).For the line to intersect the circle, this distance must be less than or equal to the radius, which is 3. So, ( |2k + 1| / sqrt{k^2 + 1} leq 3 ). Let me square both sides to eliminate the square root: ( (2k + 1)^2 / (k^2 + 1) leq 9 ). Multiply both sides by ( k^2 + 1 ): ( (4k^2 + 4k + 1) leq 9(k^2 + 1) ). Expand the right side: ( 4k^2 + 4k + 1 leq 9k^2 + 9 ). Bring everything to one side: ( -5k^2 + 4k - 8 leq 0 ). Multiply both sides by -1 (which reverses the inequality): ( 5k^2 - 4k + 8 geq 0 ).Now, the quadratic ( 5k^2 - 4k + 8 ) has discriminant ( (-4)^2 - 4*5*8 = 16 - 160 = -144 ), which is negative. Since the coefficient of ( k^2 ) is positive, the quadratic is always positive. So, the inequality ( 5k^2 - 4k + 8 geq 0 ) is always true. Therefore, the distance condition is always satisfied, meaning the line l always intersects the circle C. So, statement C is true.Finally, statement D: The shortest chord cut by circle C on line l is ( 4sqrt{2} ). Hmm, the length of the chord can be found using the formula ( 2sqrt{r^2 - d^2} ), where r is the radius and d is the distance from the center to the line. We already have r = 3, and d is ( |2k + 1| / sqrt{k^2 + 1} ).So, the chord length is ( 2sqrt{9 - ( (2k + 1)^2 / (k^2 + 1) )} ). To find the shortest chord, we need to maximize d, because as d increases, the chord length decreases.The maximum possible value of d occurs when the line is farthest from the center, but since the line passes through (-1, 1), which is inside the circle (distance from center to (-1,1) is sqrt[(1 - (-1))^2 + (0 - 1)^2] = sqrt[4 + 1] = sqrt(5) ≈ 2.236, which is less than radius 3), the line will always intersect the circle, and the maximum distance d can be is sqrt(5). Wait, is that right?Wait, actually, the maximum distance d can be is the distance from the center to the point (-1,1), because the line passes through that point. So, d can't be more than sqrt(5). Therefore, the minimum chord length is ( 2sqrt{9 - 5} = 2sqrt{4} = 4 ). But statement D says the shortest chord is ( 4sqrt{2} ), which is approximately 5.656, but we just found that the shortest chord is 4. So, statement D is false.Wait, hold on, maybe I made a mistake. Let me double-check. The chord length is ( 2sqrt{r^2 - d^2} ). If d is maximum, then chord length is minimum. The maximum d is sqrt(5), so minimum chord length is ( 2sqrt{9 - 5} = 4 ). So, 4 is the shortest chord, not ( 4sqrt{2} ). So, D is false.Therefore, statements A, B, and C are true, and D is false.

💡First, I recognize that 1000 can be expressed as 10 raised to the power of 3, since (10^3 = 1000).Next, I apply the power of a power rule, which states that ((a^m)^n = a^{m cdot n}). Applying this rule to (1000^4), I get:[(10^3)^4 = 10^{3 cdot 4} = 10^{12}]Now, the equation is simplified to (10^{12} = 10^y). Since the bases are the same, I can set the exponents equal to each other:[y = 12]Therefore, the solution is (y = 12).

💡First, I'll denote the attendance on the first day as ( x ).On the second day, due to the rain, the attendance was half of the first day, which is ( frac{x}{2} ).The third day was the finale, so the attendance tripled compared to the first day, making it ( 3x ).On the fourth day, there was a surprise guest performance, which doubled the attendance of the second day, resulting in ( 2 times frac{x}{2} = x ).Adding up the attendance for all four days:[ x + frac{x}{2} + 3x + x = 5.5x ]Given that the total attendance was 3600 people:[ 5.5x = 3600 ]Solving for ( x ):[ x = frac{3600}{5.5} approx 654.55 ]Since the number of people must be a whole number, I'll round ( x ) to the nearest whole number, which is 655.Now, calculating the attendance for each day:- First day: 655 people- Second day: ( frac{655}{2} = 327.5 ) (rounded to 328 people)- Third day: ( 3 times 655 = 1965 ) people- Fourth day: 655 peopleAdding these up:[ 655 + 328 + 1965 + 655 = 3603 ]However, this total is slightly over 3600. To adjust, I'll use the exact fractional values without rounding:- First day: ( x = 654.5454... )- Second day: ( frac{x}{2} = 327.2727... )- Third day: ( 3x = 1963.6363... )- Fourth day: ( x = 654.5454... )Adding these exact values:[ 654.5454... + 327.2727... + 1963.6363... + 654.5454... = 3600 ]Thus, the total attendance for all four days of the festival is 3600 people.

💡First, I need to determine the total number of three-digit numbers. Three-digit numbers range from 100 to 999, which gives a total of 900 numbers.Next, I'll calculate how many of these numbers are multiples of 3. The smallest three-digit multiple of 3 is 102, and the largest is 999. By dividing 999 by 3, I find there are 333 multiples of 3. Subtracting the smaller multiples below 100, there are 300 three-digit multiples of 3.Then, I'll find the number of three-digit multiples of 8. The smallest three-digit multiple of 8 is 104, and the largest is 992. Dividing 992 by 8 gives 124 multiples of 8. Subtracting the smaller multiples below 100, there are 112 three-digit multiples of 8.To avoid double-counting numbers that are multiples of both 3 and 8 (which are multiples of 24), I'll calculate these as well. The smallest three-digit multiple of 24 is 120, and the largest is 984. Dividing 984 by 24 gives 41 multiples of 24. Subtracting the smaller multiples below 100, there are 37 three-digit multiples of 24.Using the inclusion-exclusion principle, the total number of three-digit numbers that are multiples of either 3 or 8 is 300 + 112 - 37 = 375.Finally, subtracting this from the total number of three-digit numbers gives the count of numbers that are multiples of neither 3 nor 8: 900 - 375 = 525.

💡First, I need to determine the total miles Alice walks each day by adding the walking distances from her morning and evening routines.**Monday:**- Morning: 8 miles (field) + 2 miles (hill) = 10 miles- Evening: 5 miles (forest) + 6 miles (home) = 11 miles- **Total for Monday:** 10 + 11 = 21 miles**Tuesday:**- Morning: 4 miles (beach) + 3 miles (swamp) + 3 miles (uphill) = 10 miles- Evening: 2 miles (mountain slope) + 2 miles (home) = 4 miles- **Total for Tuesday:** 10 + 4 = 14 miles**Wednesday:**- Morning: 10 miles (farms and gardens)- Evening: 2 miles (forest) + 6 miles (cliff) + 4 miles (down to house) = 12 miles- **Total for Wednesday:** 10 + 12 = 22 miles**Thursday:**- Morning: 6 miles (hilly trail) + 2 miles (fort) + 2 miles (valley) = 10 miles- Evening: 3 miles (mountain path) + 6 miles (home through fields) = 9 miles- **Total for Thursday:** 10 + 9 = 19 miles**Friday:**- Morning: 5 miles (nature reserve) + 3 miles (to school) = 8 miles- Evening: 5 miles (meadow) + 2 miles (riverbank) + 5 miles (forest) = 12 miles- **Total for Friday:** 8 + 12 = 20 milesFinally, I sum up the totals for each day to find the total miles walked during the week:- **Monday:** 21 miles- **Tuesday:** 14 miles- **Wednesday:** 22 miles- **Thursday:** 19 miles- **Friday:** 20 miles**Total miles walked for the week:** 21 + 14 + 22 + 19 + 20 = 96 miles