💡Okay, so I need to show that in a quadrilateral ABCD, the diagonals AC and BD are perpendicular if and only if AB² + CD² equals BC² + AD². Hmm, let me try to break this down.First, I remember that in geometry, when we talk about perpendicular lines, their dot product is zero. Maybe I can use vectors here. Let me assign position vectors to the points A, B, C, and D. Let's say vector A is at position vector **a**, B at **b**, C at **c**, and D at **d**. So, the diagonal AC can be represented as the vector from A to C, which is **c** - **a**. Similarly, the diagonal BD is the vector from B to D, which is **d** - **b**. If these two vectors are perpendicular, their dot product should be zero. So, (**c** - **a**) · (**d** - **b**) = 0.Now, I need to relate this to the sides of the quadrilateral. The sides AB, BC, CD, and DA can be expressed as vectors too. AB is **b** - **a**, BC is **c** - **b**, CD is **d** - **c**, and DA is **a** - **d**. The problem states that AB² + CD² = BC² + AD². Let me write that in terms of vectors:|**b** - **a**|² + |**d** - **c**|² = |**c** - **b**|² + |**a** - **d**|².Expanding these magnitudes using the dot product formula |**v**|² = **v** · **v**, we get:(**b** - **a**) · (**b** - **a**) + (**d** - **c**) · (**d** - **c**) = (**c** - **b**) · (**c** - **b**) + (**a** - **d**) · (**a** - **d**).Let me compute each term:1. (**b** - **a**) · (**b** - **a**) = |**b**|² - 2**a**·**b** + |**a**|²2. (**d** - **c**) · (**d** - **c**) = |**d**|² - 2**c**·**d** + |**c**|²3. (**c** - **b**) · (**c** - **b**) = |**c**|² - 2**b**·**c** + |**b**|²4. (**a** - **d**) · (**a** - **d**) = |**a**|² - 2**a**·**d** + |**d**|²Now, let me plug these back into the equation:(|**b**|² - 2**a**·**b** + |**a**|²) + (|**d**|² - 2**c**·**d** + |**c**|²) = (|**c**|² - 2**b**·**c** + |**b**|²) + (|**a**|² - 2**a**·**d** + |**d**|²).Simplify both sides:Left side: |**a**|² + |**b**|² + |**c**|² + |**d**|² - 2**a**·**b** - 2**c**·**d**Right side: |**a**|² + |**b**|² + |**c**|² + |**d**|² - 2**b**·**c** - 2**a**·**d**Subtract the right side from the left side:(|**a**|² + |**b**|² + |**c**|² + |**d**|² - 2**a**·**b** - 2**c**·**d**) - (|**a**|² + |**b**|² + |**c**|² + |**d**|² - 2**b**·**c** - 2**a**·**d**) = 0Simplify:-2**a**·**b** - 2**c**·**d** + 2**b**·**c** + 2**a**·**d** = 0Divide both sides by 2:-**a**·**b** - **c**·**d** + **b**·**c** + **a**·**d** = 0Rearrange terms:(**a**·**d** - **a**·**b**) + (**b**·**c** - **c**·**d**) = 0Factor out **a** and **c**:**a**·(**d** - **b**) + **c**·(**b** - **d**) = 0Notice that (**b** - **d**) is the negative of (**d** - **b**), so:**a**·(**d** - **b**) - **c**·(**d** - **b**) = 0Factor out (**d** - **b**):(**a** - **c**) · (**d** - **b**) = 0But (**a** - **c**) is the vector from C to A, which is -(**c** - **a**), and (**d** - **b**) is the vector from B to D, which is (**d** - **b**). So, this simplifies to:(**c** - **a**) · (**d** - **b**) = 0Which is exactly the condition we had earlier for the diagonals AC and BD to be perpendicular. So, this shows that if AB² + CD² = BC² + AD², then the diagonals are perpendicular.Now, for the converse, if the diagonals are perpendicular, then (**c** - **a**) · (**d** - **b**) = 0. Following the same steps in reverse, we can show that AB² + CD² = BC² + AD². Therefore, the diagonals AC and BD are perpendicular if and only if AB² + CD² = BC² + AD².**Final Answer**The diagonals ( AC ) and ( BD ) are perpendicular if and only if ( AB^2 + CD^2 = BC^2 + AD^2 ). Thus, the final result is boxed{AB^2 + CD^2 = BC^2 + AD^2}.

💡Okay, so I have this problem where I need to express the margin M in terms of the selling price S. The margin is defined as M = (1/2)(S - (1/n)C), where C is the cost price. I need to find an expression for M that only involves S, not C. Let me think about how to approach this.First, I know that in business, the selling price S is usually related to the cost price C and the margin M. I remember that S = C + M, but I'm not entirely sure if that's the exact relationship here. Maybe I should confirm that. If S is the selling price, then it should cover the cost and the margin, so yes, S = C + M makes sense.So, if S = C + M, then I can express C in terms of S and M. Let me rearrange that equation: C = S - M. Okay, so now I have C expressed in terms of S and M. That might be useful because I can substitute this into the margin formula.The margin formula is M = (1/2)(S - (1/n)C). Let me substitute C = S - M into this equation. So, replacing C with (S - M), the formula becomes:M = (1/2)(S - (1/n)(S - M))Now, I need to simplify this equation. Let me first distribute the (1/n) inside the parentheses:M = (1/2)(S - (1/n)S + (1/n)M)Okay, so now I have M = (1/2)( (1 - 1/n)S + (1/n)M ). Let me write that out more clearly:M = (1/2) * [ (1 - 1/n)S + (1/n)M ]To make this easier, I can factor out the S and M terms:M = (1/2)( ( (n - 1)/n )S + (1/n)M )Hmm, that looks a bit complicated. Maybe I should multiply both sides by 2 to eliminate the fraction on the left side:2M = ( (n - 1)/n )S + (1/n)MNow, I want to get all the terms involving M on one side. Let me subtract (1/n)M from both sides:2M - (1/n)M = ( (n - 1)/n )SFactor out M on the left side:M(2 - 1/n) = ( (n - 1)/n )SNow, I can write 2 as 2n/n to have a common denominator:M( (2n - 1)/n ) = ( (n - 1)/n )SNow, to solve for M, I can multiply both sides by n/(2n - 1):M = ( (n - 1)/n )S * (n/(2n - 1))Simplify this expression:M = (n - 1)/(2n - 1) * SSo, M = (n - 1)/(2n - 1) * SLooking back at the options provided:A) (1/(n+1))SB) (n/(n+1))SC) ((n-1)/(2n - 1))SD) (n/(n-1))SIt looks like option C matches my result. So, the margin M in terms of the selling price S is ((n-1)/(2n - 1))S.Wait, let me double-check my steps to make sure I didn't make any mistakes.1. Started with M = (1/2)(S - (1/n)C)2. Expressed C as S - M3. Substituted into the equation: M = (1/2)(S - (1/n)(S - M))4. Distributed the (1/n): M = (1/2)(S - (1/n)S + (1/n)M)5. Simplified inside the parentheses: M = (1/2)(( (n - 1)/n )S + (1/n)M)6. Multiplied both sides by 2: 2M = ( (n - 1)/n )S + (1/n)M7. Subtracted (1/n)M: 2M - (1/n)M = ( (n - 1)/n )S8. Factored out M: M(2 - 1/n) = ( (n - 1)/n )S9. Expressed 2 as 2n/n: M( (2n - 1)/n ) = ( (n - 1)/n )S10. Multiplied both sides by n/(2n - 1): M = (n - 1)/(2n - 1) * SEverything seems to check out. So, I'm confident that option C is the correct answer.

💡First, I identify the points scored by each player based on the given information.Tobee scored 4 points.Jay scored 6 more than twice Tobee's points:2 * 4 + 6 = 14 points.Sean scored half of Jay's points:14 / 2 = 7 points.Remy scored 3 less than the sum of Tobee's and Jay's points:4 + 14 - 3 = 15 points.Alex scored 4 more than the sum of Sean's and Remy's points:7 + 15 + 4 = 26 points.Finally, I add up all the points scored by each player to find the team's total score:4 + 14 + 7 + 15 + 26 = 66 points.

💡Okay, so I have this problem where a program diagram results in S being equal to 35, and I need to figure out the condition about k that should be filled in the box. The options are A: (k geqslant 6), B: (k geqslant 7), C: (k geqslant 8), and D: (k > 7). First, I need to understand what the program is doing. Since it's a loop structure, I assume that the program is adding some value to S in each iteration and incrementing or decrementing k until a certain condition is met. The result S=35 suggests that the loop runs a specific number of times before exiting.Let me try to simulate the program step by step. I'll assume that in each iteration, k decreases by 1, and S increases by some value. Let's say the loop starts with k=10 and S=0. Then, in each iteration, S increases by k, and k decreases by 1. So:1. First iteration: k=10, S=0+10=10, k becomes 9.2. Second iteration: k=9, S=10+9=19, k becomes 8.3. Third iteration: k=8, S=19+8=27, k becomes 7.4. Fourth iteration: k=7, S=27+7=34, k becomes 6.5. Fifth iteration: k=6, S=34+6=40, k becomes 5.Wait, but the result is S=35, not 40. So maybe my assumption about how S is updated is incorrect. Perhaps S is being updated differently. Maybe S is being multiplied or something else.Alternatively, maybe S is being added in a different way. Let me try another approach. Suppose S starts at 0, and in each iteration, we add k to S, but only if a certain condition is met. The loop continues until k is less than a certain value.Given that S=35, let's see what combinations of k could lead to that. If k starts at 10 and decreases by 1 each time, and we add k to S until k is less than a certain value.Let's calculate the sum of k from 10 down to a certain point:- Sum from 10 to 7: 10 + 9 + 8 + 7 = 34- Sum from 10 to 6: 10 + 9 + 8 + 7 + 6 = 40But we need S=35, which is between these two sums. So maybe the loop stops before adding 6. That would mean the loop stops when k=6, so the condition is not met when k=6. Therefore, the condition should be something like k >=7, because when k=7, the condition is met, and the loop continues, but when k=6, the condition is not met, and the loop exits.Wait, let's verify that. If the condition is k >=7, then:1. k=10: condition met, add 10 to S, S=10, k=92. k=9: condition met, add 9 to S, S=19, k=83. k=8: condition met, add 8 to S, S=27, k=74. k=7: condition met, add 7 to S, S=34, k=65. k=6: condition not met, loop exitsBut S=34, not 35. Hmm, that's not matching. Maybe I missed something.Alternatively, perhaps the loop starts with k=10 and S=0, and in each iteration, S is updated by adding k, then k is decremented. The loop continues while k > some value.If the loop exits when k=6, then the last addition would be when k=7, making S=34. But we need S=35. So maybe the loop adds one more time when k=6, but then exits. That would make S=34 +6=40, which is too much.Wait, maybe the initial value of S is different. If S starts at 1 instead of 0, then:1. k=10: add 10, S=11, k=92. k=9: add 9, S=20, k=83. k=8: add 8, S=28, k=74. k=7: add 7, S=35, k=65. k=6: condition not met, loop exitsAh, that works! So if S starts at 1, and the loop adds k until k=6, then S=35 when k=6. Therefore, the condition should be k >=7, because when k=7, the condition is met, and the loop continues, adding 7 to S to make it 35, and then k becomes 6, which doesn't meet the condition, so the loop exits.Therefore, the condition is (k geqslant 7), which is option B.

💡Okay, so I have this problem where two couriers are traveling from point A to point B. The first courier takes 14 hours to go from A to B. The second courier starts at the same time but from a place that's 10 km behind A, and they both arrive at B at the same time. Additionally, the second courier covers 20 km in half an hour less than the first one. I need to find the distance from A to B.Hmm, let me try to break this down. Let's denote the distance from A to B as ( x ) kilometers. So, the first courier's speed would be ( frac{x}{14} ) km/h because they cover ( x ) km in 14 hours. Now, the second courier starts 10 km behind A, which means they have to cover ( x + 10 ) km to reach B. Since they also take 14 hours to reach B, their speed would be ( frac{x + 10}{14} ) km/h.The problem also states that the second courier covers 20 km in half an hour less than the first one. Let's figure out the time each takes to cover 20 km.For the first courier, the time to cover 20 km is ( frac{20}{frac{x}{14}} = frac{280}{x} ) hours.For the second courier, the time to cover 20 km is ( frac{20}{frac{x + 10}{14}} = frac{280}{x + 10} ) hours.According to the problem, the second courier takes half an hour less, so:[ frac{280}{x} = frac{280}{x + 10} + frac{1}{2} ]Hmm, okay, so I need to solve this equation for ( x ). Let me rewrite it:[ frac{280}{x} - frac{280}{x + 10} = frac{1}{2} ]To solve this, I'll find a common denominator for the left side. The common denominator is ( x(x + 10) ). So:[ frac{280(x + 10) - 280x}{x(x + 10)} = frac{1}{2} ]Simplify the numerator:[ 280x + 2800 - 280x = 2800 ]So, the equation becomes:[ frac{2800}{x(x + 10)} = frac{1}{2} ]Cross-multiplying:[ 2800 times 2 = x(x + 10) ][ 5600 = x^2 + 10x ]So, bringing all terms to one side:[ x^2 + 10x - 5600 = 0 ]Now, I need to solve this quadratic equation. Let me use the quadratic formula:[ x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ]Here, ( a = 1 ), ( b = 10 ), and ( c = -5600 ).Calculating the discriminant:[ b^2 - 4ac = 10^2 - 4(1)(-5600) = 100 + 22400 = 22500 ]So, the square root of 22500 is 150.Thus,[ x = frac{-10 pm 150}{2} ]This gives two solutions:1. ( x = frac{-10 + 150}{2} = frac{140}{2} = 70 )2. ( x = frac{-10 - 150}{2} = frac{-160}{2} = -80 )Since distance can't be negative, we discard -80. So, ( x = 70 ) km.Wait, let me double-check. If the distance is 70 km, then the first courier's speed is ( frac{70}{14} = 5 ) km/h. The second courier's speed is ( frac{80}{14} approx 5.714 ) km/h.Time for first courier to cover 20 km: ( frac{20}{5} = 4 ) hours.Time for second courier to cover 20 km: ( frac{20}{5.714} approx 3.5 ) hours.Difference is 0.5 hours, which is half an hour. That checks out.So, the distance from A to B is 70 km.

💡Okay, so I have this problem here: "There are 12 toads in the pond and 6 outside the pond. How many toads are there in all?" Hmm, let's see. I need to find the total number of toads, both in the pond and outside. Alright, first, I should understand what the problem is asking. It's saying there are 12 toads in the pond, which means inside the water, and 6 toads outside the pond, maybe on the bank or somewhere nearby. The question is asking for the total number of toads altogether. So, I think I need to add the number of toads in the pond to the number of toads outside the pond.Let me write that down to make it clearer. If I have 12 toads in the pond and 6 toads outside, then the total number of toads should be 12 plus 6. So, 12 + 6 equals... hmm, what's 12 plus 6? Let me count on my fingers to make sure I get it right. Starting from 12, if I add 6 more, that would be 13, 14, 15, 16, 17, 18. So, 12 plus 6 equals 18.Wait, is there another way to check this? Maybe using objects or visual aids. If I imagine 12 toads in a pond, I can picture them sitting there. Then, outside the pond, there are 6 more toads. If I combine both groups, I should have all the toads together. So, adding them up makes sense.I could also think of it in terms of place value. 12 is made up of 1 ten and 2 ones, and 6 is just 6 ones. If I add the ones together, 2 plus 6 equals 8. Then, I still have the 1 ten, so that makes 18 altogether. That seems to match what I got before.Another way to verify is by using subtraction. If I have 18 toads in total and I take away the 6 that are outside the pond, I should be left with the 12 in the pond. So, 18 minus 6 equals 12, which checks out. Similarly, if I take away the 12 in the pond from the total, I should get the 6 outside. 18 minus 12 equals 6, which also makes sense.I think I'm confident now that adding 12 and 6 gives me the correct total number of toads. So, the answer should be 18 toads in all.