💡Okay, so I have this geometry problem about a right triangle ABC with a right angle at C. The circumcenter is U, and there are points D and E on sides AC and BC respectively such that angle EUD is 90 degrees. Then, F and G are the feet of the perpendiculars from D and E to AB. I need to prove that FG is half the length of AB.First, I need to recall some properties of right triangles and circumcenters. In a right triangle, the circumcenter is the midpoint of the hypotenuse. So, in triangle ABC, since it's right-angled at C, the hypotenuse is AB, and U is the midpoint of AB. That means AU = UB = (1/2)AB.Now, points D and E are on AC and BC respectively. So, D is somewhere between A and C, and E is somewhere between B and C. The condition given is that angle EUD is 90 degrees. That seems important. Maybe I can use some properties of circles or right angles here.Since U is the circumcenter, it's equidistant from all three vertices A, B, and C. So, UA = UB = UC. That might come into play.Now, F and G are the feet of the perpendiculars from D and E to AB. So, DF is perpendicular to AB, and EG is perpendicular to AB. Therefore, DF and EG are both altitudes from D and E to the hypotenuse AB.I need to find the length of FG, which is the segment between the feet of these two perpendiculars. The goal is to show that FG is half of AB.Maybe I can use coordinate geometry to model this situation. Let me place triangle ABC on a coordinate system with point C at the origin (0,0), point A on the x-axis at (a,0), and point B on the y-axis at (0,b). Then, AB will be the hypotenuse from (a,0) to (0,b).The circumcenter U is the midpoint of AB, so its coordinates will be ((a/2), (b/2)).Now, points D and E are on AC and BC respectively. Let me parameterize their positions. Let’s say D is at some point along AC, which goes from (a,0) to (0,0). So, D can be represented as (d, 0) where 0 ≤ d ≤ a. Similarly, E is on BC, which goes from (0,b) to (0,0), so E can be represented as (0, e) where 0 ≤ e ≤ b.Given that angle EUD is 90 degrees, I can use the property that if two lines are perpendicular, the product of their slopes is -1. So, I need to find the slopes of lines UE and UD and set their product to -1.First, let's find the coordinates of U, which is ((a/2), (b/2)).The coordinates of E are (0, e) and D are (d, 0).So, the vector from U to E is (0 - a/2, e - b/2) = (-a/2, e - b/2).The vector from U to D is (d - a/2, 0 - b/2) = (d - a/2, -b/2).Since angle EUD is 90 degrees, the dot product of vectors UE and UD should be zero.So, the dot product is:(-a/2)(d - a/2) + (e - b/2)(-b/2) = 0Let me compute this:(-a/2)(d - a/2) = (-a/2)d + (a^2)/4(e - b/2)(-b/2) = (-b/2)e + (b^2)/4Adding these together:(-a/2)d + (a^2)/4 + (-b/2)e + (b^2)/4 = 0Combine like terms:(-a/2)d - (b/2)e + (a^2 + b^2)/4 = 0Multiply both sides by 4 to eliminate denominators:-2a d - 2b e + (a^2 + b^2) = 0Rearranged:2a d + 2b e = a^2 + b^2Divide both sides by 2:a d + b e = (a^2 + b^2)/2So, that's one equation relating d and e.Now, I need to find the coordinates of F and G, which are the feet of the perpendiculars from D and E to AB.First, let's find the equation of AB. Since A is (a,0) and B is (0,b), the slope of AB is (b - 0)/(0 - a) = -b/a. Therefore, the equation of AB is y = (-b/a)x + b.But wait, at x=0, y=b, which is correct, and at x=a, y=0, which is also correct.Now, the foot of the perpendicular from a point (x0, y0) to the line Ax + By + C = 0 is given by a formula, but maybe it's easier to use parametric equations or solve the system.Alternatively, since DF is perpendicular to AB, which has slope -b/a, the slope of DF is a/b.Similarly, EG is perpendicular to AB, so it also has slope a/b.Let me find the coordinates of F, the foot from D(d,0) to AB.The line DF has slope a/b, so its equation is y - 0 = (a/b)(x - d).So, y = (a/b)(x - d)Now, find the intersection point F between DF and AB.AB has equation y = (-b/a)x + b.Set them equal:(a/b)(x - d) = (-b/a)x + bMultiply both sides by ab to eliminate denominators:a^2 (x - d) = -b^2 x + ab^2Expand:a^2 x - a^2 d = -b^2 x + ab^2Bring all terms to left side:a^2 x + b^2 x - a^2 d - ab^2 = 0Factor x:x(a^2 + b^2) = a^2 d + ab^2Therefore,x = (a^2 d + ab^2)/(a^2 + b^2)Similarly, y = (a/b)(x - d) = (a/b)( (a^2 d + ab^2)/(a^2 + b^2) - d )Simplify the expression inside the parentheses:= (a^2 d + ab^2 - d(a^2 + b^2))/(a^2 + b^2)= (a^2 d + ab^2 - a^2 d - b^2 d)/(a^2 + b^2)= (ab^2 - b^2 d)/(a^2 + b^2)= b^2(a - d)/(a^2 + b^2)Therefore, y = (a/b) * [ b^2(a - d)/(a^2 + b^2) ] = a b (a - d)/(a^2 + b^2)So, coordinates of F are:( (a^2 d + ab^2)/(a^2 + b^2), a b (a - d)/(a^2 + b^2) )Similarly, let's find the coordinates of G, the foot from E(0,e) to AB.The line EG has slope a/b, so its equation is y - e = (a/b)(x - 0) => y = (a/b)x + eFind intersection G with AB: y = (-b/a)x + bSet equal:(a/b)x + e = (-b/a)x + bMultiply both sides by ab:a^2 x + ab e = -b^2 x + ab^2Bring all terms to left:a^2 x + b^2 x + ab e - ab^2 = 0Factor x:x(a^2 + b^2) = ab(b - e)Therefore,x = ab(b - e)/(a^2 + b^2)Similarly, y = (a/b)x + e = (a/b)(ab(b - e)/(a^2 + b^2)) + eSimplify:= a^2 (b - e)/(a^2 + b^2) + e= [a^2 (b - e) + e(a^2 + b^2)]/(a^2 + b^2)= [a^2 b - a^2 e + a^2 e + b^2 e]/(a^2 + b^2)= (a^2 b + b^2 e)/(a^2 + b^2)So, coordinates of G are:( ab(b - e)/(a^2 + b^2), (a^2 b + b^2 e)/(a^2 + b^2) )Now, I have coordinates for F and G. I need to find the distance between F and G, which is FG.Let me denote the coordinates:F: ( (a^2 d + ab^2)/(a^2 + b^2), a b (a - d)/(a^2 + b^2) )G: ( ab(b - e)/(a^2 + b^2), (a^2 b + b^2 e)/(a^2 + b^2) )Let me compute the differences in x and y coordinates.Δx = [ ab(b - e) - (a^2 d + ab^2) ] / (a^2 + b^2 )Simplify numerator:ab(b - e) - a^2 d - ab^2 = a b^2 - a b e - a^2 d - a b^2 = -a b e - a^2 dSo, Δx = (-a b e - a^2 d)/(a^2 + b^2 ) = -a(b e + a d)/(a^2 + b^2 )Similarly, Δy = [ (a^2 b + b^2 e) - a b (a - d) ] / (a^2 + b^2 )Simplify numerator:a^2 b + b^2 e - a^2 b + a b d = b^2 e + a b dSo, Δy = (b^2 e + a b d)/(a^2 + b^2 ) = b(b e + a d)/(a^2 + b^2 )Now, the distance FG is sqrt( (Δx)^2 + (Δy)^2 )Compute (Δx)^2 + (Δy)^2:= [ (-a(b e + a d)/(a^2 + b^2 ))^2 + (b(b e + a d)/(a^2 + b^2 ))^2 ]Factor out (b e + a d)^2 / (a^2 + b^2 )^2:= (b e + a d)^2 [ a^2 + b^2 ] / (a^2 + b^2 )^2Simplify:= (b e + a d)^2 / (a^2 + b^2 )Therefore, FG = sqrt( (b e + a d)^2 / (a^2 + b^2 ) ) = |b e + a d| / sqrt(a^2 + b^2 )Since lengths are positive, we can drop the absolute value:FG = (b e + a d)/sqrt(a^2 + b^2 )Now, from earlier, we had the equation from the perpendicular condition:a d + b e = (a^2 + b^2 ) / 2So, substituting into FG:FG = ( (a^2 + b^2 ) / 2 ) / sqrt(a^2 + b^2 ) = (a^2 + b^2 ) / (2 sqrt(a^2 + b^2 )) = sqrt(a^2 + b^2 ) / 2But sqrt(a^2 + b^2 ) is the length of AB, since AB is the hypotenuse of the right triangle ABC.Therefore, FG = AB / 2So, FG is half the length of AB, which is what we needed to prove.

💡First, I need to understand the conditions for a point to be in the second quadrant. In the coordinate plane, the second quadrant is where the x-coordinate is negative and the y-coordinate is positive.Given the point ( P(m-1, 3) ), the y-coordinate is already positive since it is 3. Therefore, I only need to ensure that the x-coordinate ( m-1 ) is negative.To find the values of ( m ) that satisfy this condition, I set up the inequality:[m - 1 < 0]Solving for ( m ), I add 1 to both sides of the inequality:[m < 1]This means that any value of ( m ) less than 1 will place the point ( P ) in the second quadrant. For simplicity, I can choose ( m = 0 ) as a possible value.Finally, substituting ( m = 0 ) into the coordinates of point ( P ), I get:[P(0 - 1, 3) = P(-1, 3)]This confirms that the point ( P(-1, 3) ) is indeed in the second quadrant.

💡Okay, so I have this geometry problem here with four statements about lines and planes. I need to figure out which ones are correct. Let me take each statement one by one and think through them carefully.Starting with statement ①: If ( m perp p ) and ( m parallel n ), then ( n perp p ). Hmm, okay. So, if line m is perpendicular to line p, and line m is parallel to line n, does that mean line n is also perpendicular to line p? I remember that if two lines are parallel, they have the same slope, and if one is perpendicular to another line, the other should be too. So, yeah, this seems correct. If m is perpendicular to p, and n is just a parallel version of m, then n should also be perpendicular to p. So, I think ① is correct.Moving on to statement ②: If ( m parallel beta ), ( n parallel beta ), ( m subset alpha ), ( n subset alpha ), then ( alpha parallel beta ). Hmm, this is a bit trickier. So, if two lines m and n are both parallel to plane β, and both lines are contained within plane α, does that mean plane α is parallel to plane β? I think for two planes to be parallel, every line in one plane should be parallel to the other plane. But wait, just having two lines in α parallel to β isn't enough. Those two lines need to be non-parallel themselves, right? Because if they're parallel, they might not span the entire plane. So, unless m and n intersect, we can't be sure that α is parallel to β. Since the problem doesn't specify that m and n intersect, I think statement ② might not always be true. So, ② is incorrect.Now, statement ③: If ( alpha perp gamma ), ( beta perp gamma ), ( alpha cap beta = m ), then ( m perp gamma ). Okay, so both planes α and β are perpendicular to plane γ, and their intersection is line m. I remember that if two planes are both perpendicular to a third plane, their line of intersection should also be perpendicular to that third plane. Let me visualize this. Imagine γ is the floor, and α and β are two walls meeting at a corner (line m). Both walls are perpendicular to the floor, so the line where they meet (the corner) is also perpendicular to the floor. Yeah, that makes sense. So, ③ is correct.Finally, statement ④: If ( alpha parallel beta ), ( m subset alpha ), ( n subset beta ), then ( m parallel n ). Hmm, so if two planes are parallel, and each contains a line, does that mean the lines are parallel? I don't think so. Because in parallel planes, lines can be parallel, but they can also be skew. For example, imagine two parallel sheets of paper. If I draw a horizontal line on one and a vertical line on the other, those lines aren't parallel; they're skew. So, just because the planes are parallel doesn't mean the lines inside them have to be parallel. They could be skew. Therefore, statement ④ is incorrect.So, after going through each statement, I think ① and ③ are correct, while ② and ④ are not. Let me just double-check to make sure I didn't miss anything.For ①, parallel lines maintain their perpendicularity to another line, so that's solid. For ②, without the lines intersecting, the planes might not necessarily be parallel. For ③, the intersection line being perpendicular makes sense with both planes being perpendicular to γ. And for ④, lines in parallel planes can be skew, so that's definitely not always true.Yeah, I feel confident now. The correct statements are ① and ③.**Final Answer**The correct statements are boxed{①③}.

💡First, I need to understand the given ratio of zinc to copper, which is 9:11. This means for every 9 parts of zinc, there are 11 parts of copper.Next, I'll calculate the total number of parts in the mixture by adding the parts of zinc and copper together: 9 parts (zinc) + 11 parts (copper) = 20 parts.The total weight of the mixture is 70 kg. To find out how much zinc is in the mixture, I'll determine the fraction of the total weight that zinc represents. Since zinc is 9 parts out of 20, the fraction is 9/20.Finally, I'll multiply this fraction by the total weight to find the weight of zinc in the mixture:Weight of zinc = (9/20) × 70 kg = 31.5 kg.

💡Okay, so I've got this problem about Juvy's garden, and I need to figure out how many chives she'll plant. Let me try to understand what's going on here.First, the garden has 20 rows, and each row has 10 plants. So, in total, there are 20 times 10 plants, which is 200 plants. That's a lot of plants! Now, Juvy is planting different herbs in these rows: parsley, rosemary, and chives.She starts by planting parsley in the first 3 rows. Since each row has 10 plants, that means she's planting 3 times 10, which is 30 parsley plants. Okay, got that.Then, she plants rosemary in the last two rows. Again, each row has 10 plants, so that's 2 times 10, which equals 20 rosemary plants. So far, she's planted 30 parsley and 20 rosemary plants.Now, the rest of the rows are going to be planted with chives. To find out how many rows are left for chives, I need to subtract the rows used for parsley and rosemary from the total number of rows. So, 20 total rows minus 3 rows for parsley and minus 2 rows for rosemary. That leaves us with 15 rows for chives.Since each of these 15 rows has 10 plants, the total number of chive plants will be 15 times 10, which is 150. So, Juvy will plant 150 chives.Wait a minute, let me double-check that. Total rows are 20. First 3 are parsley, last 2 are rosemary. So, 3 plus 2 is 5 rows accounted for. 20 minus 5 is 15 rows left for chives. Each row has 10 plants, so 15 times 10 is indeed 150. Yeah, that seems right.I think I've got it. So, Juvy will plant 150 chives in her garden.

💡Okay, so I have this problem about finding the mean salary of 100 employees in a company. The salaries are given in three different amounts: 6000, 4000, and 2500. The number of employees receiving each salary is also provided: 5 employees get 6000, 15 employees get 4000, and 80 employees get 2500. I need to find the mean salary, which is denoted as m.Alright, first, I remember that the mean is calculated by adding up all the salaries and then dividing by the total number of employees. So, I need to find the total sum of all salaries and then divide that by 100 since there are 100 employees.Let me start by calculating the total salary for each group. For the 5 employees earning 6000 each, the total would be 5 multiplied by 6000. Let me write that down: 5 * 6000 = 30,000. Okay, so that group contributes 30,000 to the total salary.Next, for the 15 employees earning 4000 each, the total would be 15 multiplied by 4000. So, 15 * 4000 = 60,000. Got it, that group adds 60,000 to the total.Now, for the largest group, which is 80 employees earning 2500 each. The total for this group would be 80 multiplied by 2500. Let me calculate that: 80 * 2500. Hmm, 80 times 2000 is 160,000, and 80 times 500 is 40,000. So, adding those together, 160,000 + 40,000 = 200,000. So, this group contributes 200,000 to the total salary.Now, I need to add up all these totals to find the overall total salary. So, adding 30,000 + 60,000 + 200,000. Let's see: 30,000 + 60,000 is 90,000, and then adding 200,000 gives me 290,000. So, the total salary for all employees combined is 290,000.Now, to find the mean salary, I need to divide this total by the number of employees, which is 100. So, m = 290,000 / 100. Dividing 290,000 by 100 is straightforward; it's just moving the decimal point two places to the left, which gives me 2,900.Wait, let me double-check my calculations to make sure I didn't make any mistakes. Starting with the first group: 5 employees * 6000 = 30,000. That seems right. Second group: 15 * 4000 = 60,000. Yep, that checks out. Third group: 80 * 2500 = 200,000. That's correct too.Adding them up: 30,000 + 60,000 = 90,000, and 90,000 + 200,000 = 290,000. Dividing by 100 gives me 2,900. Okay, I think that's correct.I guess another way to think about it is to see if the mean makes sense given the distribution of salaries. Most employees (80 out of 100) are earning the lower salary of 2500, so the mean should be closer to 2500 than to 6000. 2,900 is indeed closer to 2500, so that seems reasonable.Just to be thorough, let me recalculate each step:- 5 * 6000 = 30,000- 15 * 4000 = 60,000- 80 * 2500 = 200,000- Total = 30,000 + 60,000 + 200,000 = 290,000- Mean = 290,000 / 100 = 2,900Everything still adds up. I think I'm confident that the mean salary m is 2,900.