💡Okay, so I have this problem here about finding the coordinates of the foci of a hyperbola. The equation given is (frac{x^{2}}{3} - frac{y^{2}}{4} = 1). Hmm, I remember that hyperbolas have something to do with two separate curves that mirror each other, and they have foci which are points inside each curve. First, I need to recall the standard form of a hyperbola. I think it's either (frac{x^{2}}{a^{2}} - frac{y^{2}}{b^{2}} = 1) or (frac{y^{2}}{b^{2}} - frac{x^{2}}{a^{2}} = 1), depending on whether it opens horizontally or vertically. In this case, since the positive term is with (x^{2}), it must open left and right along the x-axis. So, the standard form is (frac{x^{2}}{a^{2}} - frac{y^{2}}{b^{2}} = 1).Comparing this to the given equation, (frac{x^{2}}{3} - frac{y^{2}}{4} = 1), I can see that (a^{2} = 3) and (b^{2} = 4). Therefore, (a = sqrt{3}) and (b = 2). Now, I need to find the foci. I remember that for hyperbolas, the distance from the center to each focus is given by (c), and it relates to (a) and (b) through the equation (c^{2} = a^{2} + b^{2}). Is that right? Wait, let me think. For ellipses, it's (c^{2} = a^{2} - b^{2}), but for hyperbolas, I think it's (c^{2} = a^{2} + b^{2}). Yeah, that sounds correct because hyperbolas have more spread out foci compared to ellipses.So, plugging in the values I have:(c^{2} = a^{2} + b^{2} = 3 + 4 = 7)Therefore, (c = sqrt{7}). Since the hyperbola opens along the x-axis, the foci are located at ((pm c, 0)). So, substituting (c = sqrt{7}), the coordinates of the foci are ((pm sqrt{7}, 0)).Let me double-check my steps to make sure I didn't make any mistakes. I identified the standard form correctly, compared (a^{2}) and (b^{2}) properly, calculated (c) using the correct formula for hyperbolas, and determined the correct coordinates based on the direction the hyperbola opens. Everything seems to add up.So, I think the foci are at ((pm sqrt{7}, 0)). Looking at the options provided:A: (±l, 0) – Hmm, "l" isn't defined here, so maybe that's a typo or something.B: (±(sqrt{7}), 0) – That's what I got.C: (±(sqrt{5}), 0) – That would be if (c^{2} = 3 + 2), but (b^{2}) is 4, not 2.D: (±4, 0) – That's way too large; (c) is (sqrt{7}), which is approximately 2.645, not 4.So, option B seems to be the correct one.

💡Alright, so I'm trying to understand this problem about partially ordered sets (posets) and chains. The statement says that for a family of sets 𝒜 = {A₁, A₂, ..., Aₜ}, the minimum number of disjoint chains needed to partition the family is equal to the maximum number of elements in an antichain subset S of P.First, I need to recall what a chain and an antichain are. A chain is a totally ordered subset, meaning every two elements are comparable. An antichain is the opposite; no two elements are comparable. So, in a poset, an antichain is a set of elements where none of them are related to each other under the partial order.The problem is essentially saying that if I have a family of sets, the smallest number of chains I need to cover all the sets (partition them into chains) is equal to the size of the largest antichain in that family. That sounds familiar; I think it's related to Dilworth's theorem. Let me recall: Dilworth's theorem states that in any finite poset, the size of the largest antichain equals the minimum number of chains needed to cover the poset. So, this seems to be a restatement of Dilworth's theorem.But to make sure I understand, let me try to think through it step by step. Suppose I have a family of sets, and I want to partition them into the fewest possible chains. Each chain is a sequence where each set is a subset of the next one, right? So, for example, if I have sets like {1}, {1,2}, {1,2,3}, that's a chain because each is contained in the next.Now, an antichain would be a collection of sets where none of them contain each other. For instance, {1}, {2}, {3} would be an antichain because none of these sets are subsets of each other.So, if I have a large antichain, say of size s, that means I have s sets where none contain each other. To cover all these s sets with chains, each chain can contain at most one of these sets because if a chain contained two of them, those two would have to be comparable, which they aren't. Therefore, I need at least s chains to cover this antichain. That suggests that the minimum number of chains needed is at least the size of the largest antichain.Conversely, if I can show that the minimum number of chains needed is not more than the size of the largest antichain, then I can conclude they are equal. This is where Dilworth's theorem comes into play, but let me try to think about it without directly invoking the theorem.Maybe I can use induction. Suppose the statement is true for all posets with fewer than t elements. Now, consider a poset with t elements. Let S be the largest antichain in this poset, with size s. I need to show that the poset can be partitioned into s chains.To do this, I can consider removing one element from S and applying the induction hypothesis. But wait, if I remove one element, the remaining poset has t-1 elements, and by the induction hypothesis, it can be partitioned into s-1 chains. But then adding back the removed element might require adding another chain, bringing the total back to s. Hmm, that seems a bit vague.Alternatively, maybe I can construct the chains directly. Start by selecting an element from S and building a chain from it. Since S is an antichain, none of the other elements in S can be in the same chain. So, each element in S must start its own chain. Then, for the remaining elements, I can try to extend these chains or form new ones as needed.But I'm not sure if this approach covers all cases. Maybe I need a more systematic way. Perhaps I can use the concept of maximal elements. In a poset, a maximal element is one that is not contained in any other element. If I can find a maximal element, it must be in some chain. But how does that help with partitioning into chains?Wait, maybe I should think about the dual of Dilworth's theorem, which relates to the minimum number of antichains needed to cover the poset. But I'm not sure if that's directly applicable here.Let me try to think of a specific example. Suppose I have the poset of subsets of {1,2,3} ordered by inclusion. The largest antichain here is the set of all 2-element subsets, which has size 3. According to the theorem, I should be able to partition the power set into 3 chains. Let's see:Chain 1: {} ⊂ {1} ⊂ {1,2} ⊂ {1,2,3}Chain 2: {} ⊂ {2} ⊂ {2,3}Chain 3: {} ⊂ {3}Wait, that's only 3 chains, but the power set has 8 elements. Did I miss something? Oh, no, actually, each chain can have multiple elements, so Chain 1 has 4 elements, Chain 2 has 3, and Chain 3 has 2. But I'm supposed to partition the entire power set into chains, not necessarily covering all elements in a single chain. So, in this case, it's possible to cover all 8 elements with 3 chains, each corresponding to a different "direction" in the subset lattice.But in this example, the largest antichain is size 3, and indeed, the minimum number of chains needed is 3. So, it seems to hold here.Another example: consider the poset of divisors of 6, ordered by divisibility. The elements are {1, 2, 3, 6}. The largest antichain is {2, 3}, size 2. Can I partition this poset into 2 chains? Yes:Chain 1: 1 ⊂ 2 ⊂ 6Chain 2: 1 ⊂ 3 ⊂ 6Wait, but 1 is in both chains. That's not allowed because the chains need to be disjoint. So, I need to adjust. Maybe:Chain 1: 1 ⊂ 2 ⊂ 6Chain 2: 3 ⊂ 6But then 6 is in both chains, which is also not allowed. Hmm, so maybe I need to find a different partition. Alternatively, perhaps:Chain 1: 1 ⊂ 2Chain 2: 1 ⊂ 3Chain 3: 6But that's 3 chains, which is more than the size of the largest antichain. So, something's wrong here. Wait, maybe I'm misunderstanding the poset. In the divisor poset, 6 is the maximum element, so it can be in only one chain. Similarly, 1 is the minimum element, so it has to be in every chain? No, that can't be because chains are disjoint. So, perhaps I need to think differently.Actually, in this case, the poset is a lattice, and it's a distributive lattice, so Dilworth's theorem should apply. The largest antichain is size 2, so it should be possible to partition into 2 chains. Let's try:Chain 1: 1 ⊂ 2 ⊂ 6Chain 2: 3 ⊂ 6But as before, 6 is in both chains. That's not allowed. So, maybe:Chain 1: 1 ⊂ 2Chain 2: 1 ⊂ 3Chain 3: 6But that's 3 chains, which contradicts the theorem. Wait, maybe I'm missing something. Perhaps 6 can be in only one chain, but then how do I cover it? Maybe:Chain 1: 1 ⊂ 2 ⊂ 6Chain 2: 3But then 3 is only in a chain by itself, which is allowed, but then the chains are not necessarily required to cover all elements? No, the theorem says to partition the family, so all elements must be covered.Wait, perhaps I'm misapplying the theorem. In the divisor poset of 6, the elements are {1, 2, 3, 6}. The largest antichain is indeed {2, 3}, size 2. So, according to Dilworth's theorem, it should be possible to partition the poset into 2 chains. Let's see:Chain 1: 1 ⊂ 2 ⊂ 6Chain 2: 3 ⊂ 6But again, 6 is in both chains. That's not allowed. So, maybe the chains can share elements? No, chains must be disjoint. Hmm, this is confusing.Wait, perhaps the chains don't have to be maximal. Maybe I can have:Chain 1: 1 ⊂ 2Chain 2: 1 ⊂ 3Chain 3: 6But that's 3 chains, which is more than the size of the largest antichain. So, perhaps Dilworth's theorem doesn't apply here? Or maybe I'm misunderstanding the poset.Wait, no, Dilworth's theorem should apply to any finite poset. So, maybe I'm not partitioning correctly. Let me think again. The poset is {1, 2, 3, 6} with divisibility. The chains must be disjoint and cover all elements.One way is:Chain 1: 1 ⊂ 2 ⊂ 6Chain 2: 3But then 6 is only in Chain 1, and 3 is in Chain 2. So, all elements are covered, and the chains are disjoint. So, that's 2 chains: Chain 1 has 1, 2, 6; Chain 2 has 3. But the size of the largest antichain is 2, and we've used 2 chains. So, it works.Wait, but in this case, Chain 2 only has one element, which is allowed. So, the minimum number of chains is indeed equal to the size of the largest antichain.Okay, so that example works. Maybe I was overcomplicating it before.Another example: consider the poset of subsets of {1,2}, which has elements {}, {1}, {2}, {1,2}. The largest antichain is the set of singletons {1}, {2}, size 2. Can I partition this poset into 2 chains?Yes:Chain 1: {} ⊂ {1} ⊂ {1,2}Chain 2: {2}But again, Chain 2 only has one element. Alternatively:Chain 1: {} ⊂ {2} ⊂ {1,2}Chain 2: {1}Either way, it's 2 chains, matching the size of the largest antichain.So, these examples seem to support the theorem.Now, going back to the original problem, it's essentially stating Dilworth's theorem. So, to prove it, I can either use Dilworth's theorem directly or provide a proof of it.Since the user is asking for a step-by-step explanation, I'll try to outline a proof of Dilworth's theorem, which will show that the minimum number of chains needed to partition the poset equals the size of the largest antichain.Here's how I think the proof goes:1. **Upper Bound:** Show that the minimum number of chains needed is at least the size of the largest antichain. This is because each chain can contain at most one element from the antichain, so you need at least as many chains as the size of the antichain.2. **Lower Bound:** Show that the minimum number of chains needed is at most the size of the largest antichain. This involves constructing a partition into that many chains.To establish the lower bound, one common approach is to use induction on the size of the poset. Here's a rough sketch:- **Base Case:** If the poset has one element, the largest antichain is 1, and obviously, you need one chain.- **Inductive Step:** Assume the theorem holds for all posets with fewer than n elements. Now, consider a poset P with n elements. Let S be a largest antichain in P. Remove one element from S, say x, and apply the induction hypothesis to the remaining poset P {x}. By the induction hypothesis, P {x} can be partitioned into k chains, where k is the size of the largest antichain in P {x}. Since S was a largest antichain in P, the largest antichain in P {x} is at most k. But actually, removing x might decrease the size of the largest antichain by at most 1, so the largest antichain in P {x} is at least k - 1. However, we need to be careful here.Alternatively, another approach is to use the concept of maximal chains and ensure that each element is included in some chain without overlapping with others.But I think the standard proof uses the concept of matching in bipartite graphs, but that might be more advanced.Alternatively, here's a more constructive approach:- Let S be a largest antichain in P.- For each element x in S, start a new chain with x.- For the remaining elements not in S, since they are comparable to some element in S (otherwise S wouldn't be maximal), we can extend the chains starting from S.- Since each element not in S is comparable to some element in S, we can add them to the appropriate chain.- This way, we can cover all elements with |S| chains.But I need to make sure that this process works without conflicts, i.e., that each element can be added to exactly one chain.Wait, but how do we ensure that? Maybe by processing the elements in a certain order, like starting from the minimal elements and moving up.Alternatively, perhaps using the fact that in a finite poset, every element is either in S or comparable to some element in S.Yes, that's a key point. If S is a maximal antichain, then every element not in S is comparable to at least one element in S. Therefore, we can assign each element not in S to a chain starting from some element in S.But we need to ensure that the assignment doesn't cause overlaps, i.e., that each element is assigned to only one chain.This might require some kind of greedy algorithm or matching.Alternatively, perhaps using the concept of a graded poset, where elements are assigned levels, and chains are built level by level.But I'm not sure if that's necessary.Wait, maybe I can think of it as follows:- Start with the largest antichain S.- For each element in S, create a chain starting at that element.- Then, for each element not in S, find the minimal element in S that it is comparable to and add it to that chain.- Since S is an antichain, no two elements in S are comparable, so each element not in S can be added to exactly one chain.But I need to verify that this works.Suppose x is an element not in S. Since S is a maximal antichain, x must be comparable to some element in S. Let y be the minimal element in S such that x is comparable to y. Then, we can add x to the chain starting at y.But we need to ensure that this process doesn't create any conflicts, i.e., that x isn't added to multiple chains.Since y is the minimal element in S comparable to x, and S is an antichain, y is unique in S that x is comparable to. Wait, is that true? Suppose x is comparable to multiple elements in S. For example, x could be greater than two elements in S. But since S is an antichain, those two elements are incomparable, so x being greater than both would mean that x is an upper bound for both. But in that case, x could be added to either chain, but we need to choose one.To avoid conflicts, we need a systematic way to assign x to one chain. Perhaps choosing the minimal element in S that x is comparable to ensures uniqueness.But I'm not entirely sure. Maybe I need to use a more formal approach.Alternatively, perhaps I can use the concept of a greedy algorithm: process the elements in some order, and for each element, add it to the first chain it can fit into.But to make this work, I need to ensure that the chains are built in a way that allows this.Alternatively, perhaps using the concept of a partition into chains by considering the levels of elements.Wait, maybe I'm overcomplicating it. Let me try to outline the proof more formally.**Proof:**Let P be a finite poset, and let s be the size of the largest antichain in P. We need to show that P can be partitioned into s chains.**Step 1:** Let S be a largest antichain in P, with |S| = s.**Step 2:** For each element x in S, define a chain C_x starting at x.**Step 3:** For each element y not in S, since S is a maximal antichain, y must be comparable to some element in S. Let z be the minimal element in S such that y is comparable to z. Add y to the chain C_z.**Step 4:** Since S is an antichain, for any y not in S, there is a unique minimal z in S such that y is comparable to z. Therefore, y can be uniquely assigned to C_z.**Step 5:** Thus, all elements are assigned to one of the s chains C_x, where x ranges over S.**Step 6:** Therefore, P can be partitioned into s chains, which is the size of the largest antichain.This completes the proof.Wait, but in Step 4, I assumed that for each y not in S, there is a unique minimal z in S such that y is comparable to z. Is that always true?Suppose y is comparable to multiple elements in S. Since S is an antichain, those elements are incomparable. Therefore, y could be greater than some elements in S and less than others. But since we're considering minimal elements, if y is greater than some elements in S, there might be a minimal one. Similarly, if y is less than some elements in S, there might be a maximal one.But in general, for y not in S, it's either greater than some elements in S or less than some elements in S, but not necessarily both. Wait, no, y could be both greater than some and less than others, but since S is an antichain, y can't be both greater than and less than the same element in S.But in any case, to ensure uniqueness, we need to define a way to assign y to exactly one chain. Perhaps by choosing the minimal element in S that y is greater than or the maximal element in S that y is less than, depending on the situation.Alternatively, perhaps using a linear extension of the poset to order the elements and then assign each element to the first chain it can fit into.But I think the key idea is that since S is a maximal antichain, every element not in S is comparable to some element in S, and thus can be assigned to a chain starting from that element.Therefore, the minimum number of chains needed to partition P is equal to the size of the largest antichain.So, in conclusion, the problem is essentially stating Dilworth's theorem, and the proof involves showing that the minimum number of chains needed is both at least and at most the size of the largest antichain, thereby establishing equality.

💡Okay, so I have this problem here: Given the equation ((x^2 + 1)(2x + 1)^9 = a_0 + a_1(x + 2) + a_2(x + 2)^2 + ldots + a_{11}(x + 2)^{11}), I need to find the value of (a_0 + a_1 + a_2 + ldots + a_{11}). The options are A: 2, B: -1, C: -2, D: 1.Hmm, I remember that when you have a polynomial expressed in terms of another polynomial, like here it's expressed in terms of ((x + 2)), the coefficients (a_0, a_1, ldots, a_{11}) are the coefficients when you expand the left-hand side in terms of ((x + 2)). But I need the sum of these coefficients. I think there's a trick to find the sum of coefficients in a polynomial. Isn't it just evaluating the polynomial at a specific value? Let me recall. If I have a polynomial (P(x) = a_0 + a_1x + a_2x^2 + ldots + a_nx^n), then the sum of the coefficients is (P(1)). Because plugging in 1 for x gives (a_0 + a_1 + a_2 + ldots + a_n).But in this case, the polynomial is expressed in terms of ((x + 2)), not just (x). So, does that mean I need to evaluate the polynomial at (x = -1) because ((x + 2)) becomes 1 when (x = -1)? Let me check that.If I let (x = -1), then each term ((x + 2)) becomes ((-1 + 2) = 1). So the right-hand side becomes (a_0 + a_1(1) + a_2(1)^2 + ldots + a_{11}(1)^{11}), which is just (a_0 + a_1 + a_2 + ldots + a_{11}), exactly the sum I need.So, I can substitute (x = -1) into the left-hand side as well and set it equal to the sum of the coefficients. Let's compute the left-hand side at (x = -1):First, compute (x^2 + 1) when (x = -1):[(-1)^2 + 1 = 1 + 1 = 2]Next, compute ((2x + 1)^9) when (x = -1):[2(-1) + 1 = -2 + 1 = -1]So, ((-1)^9 = -1).Now, multiply these two results together:[2 times (-1) = -2]Therefore, the sum (a_0 + a_1 + a_2 + ldots + a_{11}) is (-2).Wait, let me make sure I didn't make a mistake. I substituted (x = -1) because that makes each ((x + 2)) term equal to 1, which simplifies the right-hand side to the sum of the coefficients. Then, I computed the left-hand side at (x = -1) correctly: (x^2 + 1) gives 2, and ((2x + 1)^9) gives (-1), so the product is indeed (-2). Is there another way to verify this? Maybe by expanding the left-hand side and comparing coefficients, but that seems complicated. Alternatively, I can think about the substitution method again. Since the right-hand side is a polynomial in ((x + 2)), evaluating at (x = -1) gives the sum of the coefficients because each ((x + 2)) becomes 1, so the entire expression becomes (a_0 + a_1 + a_2 + ldots + a_{11}).Yes, that makes sense. So, I think my answer is correct.**Final Answer**boxed{C}

💡First, I need to calculate the discounted prices for each item.For the **Lawn Chair**:- The original price is 79.95.- The first discount of 20% is calculated as 0.20 * 79.95 = 15.99.- The price after the first discount is 79.95 - 15.99 = 63.96.- The second discount of 15% is calculated as 0.15 * 63.96 = 9.594, which rounds to 9.59.- The final price after both discounts is 63.96 - 9.59 = 54.37.For the **Outdoor Umbrella**:- The original price is 125.50.- The discount of 25% is calculated as 0.25 * 125.50 = 31.375, which rounds to 31.38.- The discounted price is 125.50 - 31.38 = 94.12.For the **Patio Table**:- The original price is 240.65.- There is no discount, so the price remains 240.65.Next, I calculate the sales tax for the outdoor umbrella and the patio table.For the **Sales Tax on the Outdoor Umbrella**:- The sales tax is 7% of 94.12, which is 0.07 * 94.12 = 6.5884, rounding to 6.59.For the **Sales Tax on the Patio Table**:- The sales tax is 7% of 240.65, which is 0.07 * 240.65 = 16.8455, rounding to 16.85.Now, I add up the total cost of all three items, including the sales tax for the umbrella and table.Total cost = Price of lawn chair + Price of umbrella + Sales tax for umbrella + Price of patio table + Sales tax for patio tableTotal cost = 54.37 + 94.12 + 6.59 + 240.65 + 16.85Total cost = 412.58To find the percent decrease from the original prices, I need to calculate the original total price and compare it to the discounted total price.Original total price = Price of lawn chair + Price of umbrella + Price of patio tableOriginal total price = 79.95 + 125.50 + 240.65Original total price = 446.10Percent decrease = [(Original total price - Total cost) / Original total price] * 100Percent decrease = [(446.10 - 412.58) / 446.10] * 100Percent decrease = (33.52 / 446.10) * 100Percent decrease ≈ 7.51%So, the total cost of all three items is 412.58, and the total price decreased by approximately 7.51% from the original prices.

💡Okay, so I have this geometry problem here about triangle PQR. Let me try to visualize it first. We know that side PQ is 3 units long. From vertex P, there's a median PM which is √14 units long, and an altitude PH which is √5 units long. Both of these are drawn to side QR. The problem also mentions that the sum of angles QPR and PRQ is less than 90 degrees, and we need to find the length of side PR.Hmm, let me break this down step by step. First, let's recall what a median and an altitude are. A median connects a vertex to the midpoint of the opposite side, so M must be the midpoint of QR. An altitude is a perpendicular segment from a vertex to the opposite side, so PH is perpendicular to QR.Since PH is an altitude, triangle PHQ and triangle PHR are both right-angled triangles. Maybe I can use the Pythagorean theorem here. Let me try that.Starting with triangle PHQ, which is right-angled at H. We know PQ is 3 and PH is √5. So, applying the Pythagorean theorem:PQ² = PH² + HQ²Plugging in the known values:3² = (√5)² + HQ²9 = 5 + HQ²So, HQ² = 9 - 5 = 4Therefore, HQ = √4 = 2Okay, so the length from H to Q is 2 units. Now, since M is the midpoint of QR, and PM is a median, maybe I can find the length from M to H.We know PM is √14. Since PM is the median, and PH is the altitude, they both start from P but go to different points on QR. Let me consider triangle PMH, which is also right-angled at H because PH is perpendicular to QR.So, in triangle PMH:PM² = PH² + MH²Plugging in the known values:(√14)² = (√5)² + MH²14 = 5 + MH²So, MH² = 14 - 5 = 9Therefore, MH = √9 = 3Alright, so the distance from M to H is 3 units. Now, since H is the foot of the altitude from P to QR, and M is the midpoint, I need to figure out where exactly H is located relative to M.Given that angle QPR + angle PRQ < 90 degrees, which suggests that triangle PQR is such that the sum of these two angles is acute. This might imply that the triangle is obtuse or has some specific orientation. Maybe this affects the position of H relative to M.Wait, if the sum of angles QPR and PRQ is less than 90 degrees, then angle PQR must be greater than 90 degrees because the sum of all three angles in a triangle is 180 degrees. So, angle PQR is obtuse. That means that the foot of the altitude from P, which is H, lies outside the segment QR. So, H is not between Q and R but extended beyond Q.So, if H is beyond Q, then the length from Q to H is 2 units, as we found earlier. Then, since M is the midpoint of QR, the length from Q to M is half of QR. But we need to find QR first.Wait, let's think about this. If H is beyond Q, then the distance from M to H would be the distance from M to Q plus the distance from Q to H. But earlier, we found MH = 3 and HQ = 2. So, if H is beyond Q, then MH = MQ + QH.But wait, M is the midpoint of QR, so MQ = MR = (QR)/2.But if H is beyond Q, then MH = MQ + QH.So, MH = MQ + QHWe have MH = 3, QH = 2, so MQ = MH - QH = 3 - 2 = 1Therefore, MQ = 1, which means QR = 2 * MQ = 2 * 1 = 2Wait, that can't be right because if QR is 2, then the length from Q to H is 2, making H coincide with R? That doesn't make sense because H is supposed to be the foot of the altitude beyond Q.Wait, maybe I made a mistake in the direction. Let me draw this out mentally.If H is beyond Q, then QR is from Q to R, and H is beyond Q, so the order is H, Q, R. So, the distance from H to Q is 2, and from Q to R is something else. M is the midpoint of QR, so M is halfway between Q and R.So, the distance from M to H would be the distance from M to Q plus the distance from Q to H.So, MH = MQ + QHWe have MH = 3, QH = 2, so MQ = 3 - 2 = 1Therefore, QR = 2 * MQ = 2 * 1 = 2But then, if QR is 2, and QH is 2, that would mean H is at the same point as R, which contradicts the fact that H is beyond Q. So, maybe my initial assumption about the position of H is wrong.Alternatively, perhaps H is between Q and R. Let me reconsider.If H is between Q and R, then the distance from Q to H is 2, and M is the midpoint of QR. So, the distance from Q to M is (QR)/2.Then, the distance from M to H would be |MQ - QH|, depending on whether H is closer to Q or R than M.But since we found MH = 3, and QH = 2, let's see.If H is between Q and M, then MH = MQ - QHBut MQ = QR/2, so MH = QR/2 - 2 = 3So, QR/2 = 3 + 2 = 5Therefore, QR = 10Alternatively, if H is between M and R, then MH = QH + MQBut that would mean 3 = 2 + MQ, so MQ = 1, which would make QR = 2But earlier, that led to a contradiction because H would coincide with R.Wait, maybe I need to clarify the exact position of H.Given that angle QPR + angle PRQ < 90 degrees, which suggests that angle at P is such that the triangle is acute at P, but the sum of angles at Q and R is less than 90, making angle at Q or R obtuse.Wait, no, the sum of angles QPR and PRQ is less than 90, so the remaining angle at Q must be greater than 90 degrees because the total sum is 180.So, angle PQR > 90 degrees, making triangle PQR obtuse at Q.Therefore, the foot of the altitude from P, which is H, must lie outside the segment QR, beyond Q.So, H is beyond Q, meaning the order is H, Q, R.Therefore, the distance from H to Q is 2, and from Q to R is something else.Since M is the midpoint of QR, the distance from Q to M is QR/2.Now, the distance from M to H would be the distance from M to Q plus the distance from Q to H.So, MH = MQ + QHWe have MH = 3, QH = 2, so MQ = 3 - 2 = 1Therefore, QR = 2 * MQ = 2 * 1 = 2Wait, but if QR is 2, and QH is 2, then H is at the same point as R, which is not possible because H is supposed to be beyond Q.This seems contradictory. Maybe I made a mistake in the assumption.Alternatively, perhaps I need to consider that H is beyond R instead of beyond Q. Let me think.If angle PQR is obtuse, then the foot of the altitude from P would be outside QR, but depending on the triangle, it could be beyond Q or beyond R.Wait, if angle PQR is obtuse, then the foot of the altitude from P would be outside QR, but on the side opposite to the obtuse angle.Wait, actually, in a triangle, the foot of the altitude from a vertex opposite an obtuse angle lies outside the triangle.So, if angle PQR is obtuse, then the foot of the altitude from P would lie outside QR, beyond Q.So, H is beyond Q.Therefore, the distance from H to Q is 2, and from Q to R is QR.Since M is the midpoint of QR, MQ = QR/2.Then, the distance from M to H is MQ + QH = QR/2 + 2 = 3So, QR/2 + 2 = 3Therefore, QR/2 = 1So, QR = 2But then, if QR is 2, and QH is 2, that would mean H is at the same point as R, which is not possible.Hmm, this is confusing. Maybe I need to approach this differently.Let me denote QR as x. Then, since M is the midpoint, MQ = MR = x/2.We know that PH is the altitude, so PH = √5, and PQ = 3.From triangle PHQ, we have:PQ² = PH² + HQ²So, 3² = (√5)² + HQ²9 = 5 + HQ²So, HQ² = 4, hence HQ = 2Now, since H is beyond Q, the distance from H to Q is 2, and from Q to R is x.Therefore, the distance from H to R is x + 2.Now, considering triangle PHR, which is right-angled at H, we can write:PR² = PH² + HR²So, PR² = (√5)² + (x + 2)²PR² = 5 + (x + 2)²Now, we also have the median PM = √14.Since M is the midpoint of QR, we can use the formula for the length of a median in terms of the sides of the triangle.The formula is:PM² = (2PQ² + 2PR² - QR²)/4Plugging in the known values:(√14)² = (2*3² + 2PR² - x²)/414 = (18 + 2PR² - x²)/4Multiplying both sides by 4:56 = 18 + 2PR² - x²So, 2PR² - x² = 56 - 18 = 38Therefore, 2PR² - x² = 38But we also have from triangle PHR:PR² = 5 + (x + 2)²So, PR² = 5 + x² + 4x + 4 = x² + 4x + 9Plugging this into the previous equation:2(x² + 4x + 9) - x² = 382x² + 8x + 18 - x² = 38x² + 8x + 18 = 38x² + 8x - 20 = 0Now, solving this quadratic equation:x = [-8 ± √(64 + 80)] / 2 = [-8 ± √144]/2 = [-8 ± 12]/2So, x = (4)/2 = 2 or x = (-20)/2 = -10Since length cannot be negative, x = 2So, QR = 2But earlier, we saw that if QR = 2, then H would coincide with R, which is not possible because H is beyond Q.Wait, maybe I made a mistake in the sign when calculating HR.Wait, if H is beyond Q, then HR = QR + QH = x + 2But if QR = 2, then HR = 2 + 2 = 4So, PR² = 5 + 4² = 5 + 16 = 21Therefore, PR = √21Wait, but earlier, when I tried to find QR, I got QR = 2, which led to H coinciding with R, but maybe that's acceptable because H is beyond Q, so even if QR is 2, H is 2 units beyond Q, making HR = 4.So, PR² = 5 + 16 = 21, so PR = √21Therefore, the length of PR is √21.I think that makes sense. Let me just verify the steps again.1. From triangle PHQ, found HQ = 22. From triangle PMH, found MH = 33. Since H is beyond Q, MH = MQ + QH => 3 = MQ + 2 => MQ = 14. Therefore, QR = 2 * MQ = 25. Then, HR = QR + QH = 2 + 2 = 46. From triangle PHR, PR² = 5 + 16 = 21 => PR = √21Yes, that seems consistent. So, the length of PR is √21.

💡Alright, so I have this problem about repeating decimals, and I need to figure out which expression is incorrect. Let me try to understand the problem step by step.First, the problem defines a repeating decimal ( D ). It says that ( P ) represents the ( r ) figures (digits) of ( D ) that do not repeat, and ( Q ) represents the ( s ) figures that do repeat. So, for example, if ( D ) is 0.123454545..., then ( P ) would be 123 and ( Q ) would be 45, with ( r = 3 ) and ( s = 2 ).The question then gives five expressions labeled (A) through (E) and asks which one is incorrect. Let me go through each expression one by one.**Expression (A):** ( D = .PQQQldots )This seems straightforward. It's saying that the decimal ( D ) starts with the non-repeating part ( P ) and then has the repeating part ( Q ) indefinitely. That makes sense. For example, if ( P = 123 ) and ( Q = 45 ), then ( D = 0.123454545ldots ). So, this expression is correct.**Expression (B):** ( 10^rD = P.QQQldots )Multiplying ( D ) by ( 10^r ) shifts the decimal point ( r ) places to the right. So, the non-repeating part ( P ) moves to the left of the decimal, and the repeating part ( Q ) remains after the decimal. Using the same example, if ( D = 0.123454545ldots ) and ( r = 3 ), then ( 10^3D = 123.454545ldots ), which is ( P.QQQldots ). This also looks correct.**Expression (C):** ( 10^{r + s}D = PQ.QQQldots )This expression involves multiplying ( D ) by ( 10^{r + s} ). This shifts the decimal point ( r + s ) places to the right. So, both ( P ) and ( Q ) are moved to the left of the decimal, and then the repeating part ( Q ) starts again. In my example, ( r = 3 ) and ( s = 2 ), so ( 10^{5}D = 12345.454545ldots ), which is ( PQ.QQQldots ). This seems correct as well.**Expression (D):** ( 10^r(10^s - 1)D = Q(P - 1) )Okay, this one looks a bit more complicated. Let's break it down. First, ( 10^rD = P.QQQldots ) from expression (B). Then, ( 10^{r + s}D = PQ.QQQldots ) from expression (C). If I subtract these two equations, I get:( 10^{r + s}D - 10^rD = PQ.QQQldots - P.QQQldots )Simplifying the right side, the repeating parts cancel out, leaving ( PQ - P ). On the left side, factoring out ( 10^rD ), we get:( 10^rD(10^s - 1) = PQ - P )Which simplifies to:( 10^r(10^s - 1)D = P(Q - 1) )Wait, but the expression given in (D) is ( Q(P - 1) ). That doesn't match what I derived. I have ( P(Q - 1) ) instead. So, unless ( P(Q - 1) = Q(P - 1) ), which would mean ( PQ - P = QP - Q ), implying ( -P = -Q ) or ( P = Q ). But ( P ) and ( Q ) are different parts of the decimal, so they don't have to be equal. Therefore, this expression seems incorrect.**Expression (E):** ( 10^r cdot 10^{2s}D = PQQ.QQQldots )This expression involves multiplying ( D ) by ( 10^r ) and then by ( 10^{2s} ), which is the same as multiplying by ( 10^{r + 2s} ). Shifting the decimal point ( r + 2s ) places to the right, we get ( P ), followed by two instances of ( Q ), and then the repeating ( Q ). In my example, ( r = 3 ) and ( s = 2 ), so ( 10^{7}D = 1234545.454545ldots ), which is ( PQQ.QQQldots ). This seems correct.So, after analyzing each expression, it appears that expression (D) is incorrect because it doesn't match the result I derived when subtracting the two previous expressions. The correct expression should involve ( P(Q - 1) ) instead of ( Q(P - 1) ).**Final Answer**boxed{D}