answer：Okay, so I have this problem where I'm given two vectors, a and b. Vector a is (3,1) and vector b is (2,2). I need to find the cosine of the angle between the vectors a + b and a - b. The options are given as A: 1/17, B: sqrt(17)/17, C: sqrt(5)/5, and D: 2sqrt(5)/5.Alright, let me start by recalling how to find the cosine of the angle between two vectors. I remember that the formula involves the dot product of the two vectors divided by the product of their magnitudes. So, cos(theta) = (u · v) / (|u| |v|). That seems right.First, I need to find the vectors a + b and a - b. Let me compute those. Vector a is (3,1) and vector b is (2,2). So, adding them together, a + b would be (3 + 2, 1 + 2) which is (5,3). Similarly, subtracting, a - b would be (3 - 2, 1 - 2) which is (1,-1). Okay, that seems straightforward.Now, I need to find the dot product of a + b and a - b. The dot product is calculated by multiplying corresponding components and then adding them up. So, (5,3) · (1,-1) would be 5*1 + 3*(-1) = 5 - 3 = 2. Hmm, that's a small number, but okay.Next, I need the magnitudes of both a + b and a - b. The magnitude of a vector (x,y) is sqrt(x^2 + y^2). So, for a + b which is (5,3), the magnitude is sqrt(5^2 + 3^2) = sqrt(25 + 9) = sqrt(34). For a - b which is (1,-1), the magnitude is sqrt(1^2 + (-1)^2) = sqrt(1 + 1) = sqrt(2). So now, putting it all together, the cosine of the angle is the dot product divided by the product of the magnitudes. That would be 2 / (sqrt(34) * sqrt(2)). Let me compute that denominator. sqrt(34) times sqrt(2) is sqrt(34*2) which is sqrt(68). So, 2 / sqrt(68). Wait, sqrt(68) can be simplified. 68 is 4*17, so sqrt(68) is sqrt(4*17) which is 2*sqrt(17). So, 2 / (2*sqrt(17)) simplifies to 1 / sqrt(17). But usually, we rationalize the denominator, so multiplying numerator and denominator by sqrt(17), we get sqrt(17)/17. Looking back at the options, that's option B: sqrt(17)/17. So, I think that's the answer. Let me just double-check my steps to make sure I didn't make any mistakes.1. Adding vectors: (3+2,1+2) = (5,3) – correct.2. Subtracting vectors: (3-2,1-2) = (1,-1) – correct.3. Dot product: 5*1 + 3*(-1) = 5 - 3 = 2 – correct.4. Magnitudes: sqrt(25 + 9) = sqrt(34) and sqrt(1 + 1) = sqrt(2) – correct.5. Dot product over product of magnitudes: 2 / (sqrt(34)*sqrt(2)) = 2 / sqrt(68) = 2 / (2*sqrt(17)) = 1 / sqrt(17) = sqrt(17)/17 – correct.Yeah, everything seems to check out. So, I'm confident that the answer is B: sqrt(17)/17.**Final Answer**boxed{B}

answer：Okay, so I have this problem about a tetrahedron (ABCD). Let me try to visualize it first. A tetrahedron has four triangular faces, and in this case, we're given some specific measurements and angles between edges. First, the problem states that (AB = 1) and (CD = sqrt{3}). The distance between the lines (AB) and (CD) is 2, and the angle between these two lines is (frac{pi}{3}). I need to find the volume of this tetrahedron.Hmm, I remember that the volume of a tetrahedron can be calculated using the formula involving the base area and the height. But in this case, since we're dealing with two skew lines (lines that don't intersect and aren't parallel), the distance between them and the angle between them might be useful.Let me recall the formula for the volume of a tetrahedron when two skew lines are involved. I think it's something like:[text{Volume} = frac{1}{6} times |AB| times |CD| times d times sin(theta)]Where (d) is the distance between the lines and (theta) is the angle between them. Let me check if this makes sense. So, substituting the given values:[text{Volume} = frac{1}{6} times 1 times sqrt{3} times 2 times sinleft(frac{pi}{3}right)]Calculating step by step:1. (|AB| = 1)2. (|CD| = sqrt{3})3. (d = 2)4. (theta = frac{pi}{3}), so (sinleft(frac{pi}{3}right) = frac{sqrt{3}}{2})Putting it all together:[text{Volume} = frac{1}{6} times 1 times sqrt{3} times 2 times frac{sqrt{3}}{2}]Simplify the expression:- Multiply (sqrt{3}) and (frac{sqrt{3}}{2}): (sqrt{3} times frac{sqrt{3}}{2} = frac{3}{2})- Then multiply by 2: (frac{3}{2} times 2 = 3)- Now, multiply by (frac{1}{6}): (frac{1}{6} times 3 = frac{1}{2})So, the volume is (frac{1}{2}). Let me see if this matches one of the answer choices. Yes, option B is (frac{1}{2}).Wait, just to make sure I didn't make a mistake. Let me think again. The formula I used is for the volume between two skew lines, right? So, it's like the scalar triple product formula, but adapted for skew lines. I think that's correct because the volume can be thought of as the area of a base times height, and in this case, the base is related to the parallelogram formed by the vectors of the lines, and the height is the distance between them.Alternatively, maybe I can think of it as the area of one line segment times the height from the other line. But I think the formula I used is the standard one for this scenario.Let me double-check the formula. I found a reference that says the volume of a tetrahedron formed by two skew lines can be calculated as (frac{1}{6}) times the product of the lengths of the lines, the distance between them, and the sine of the angle between them. So, yes, that seems right.Therefore, I'm confident that the volume is (frac{1}{2}), which is option B.

answer：Okay, so I have this problem about a square ABCD with a side length of 20 cm. Points E and F are the midpoints of sides AB and BC, respectively. I need to find the area of quadrilateral BEGF. Hmm, let me try to visualize this first.Alright, so square ABCD. Let me label the square: A is the top-left corner, B is the top-right, C is the bottom-right, and D is the bottom-left. So, AB is the top side, BC is the right side, CD is the bottom side, and DA is the left side.Points E and F are midpoints. So, E is the midpoint of AB, which means it's halfway along the top side. Since AB is 20 cm, E divides it into two segments of 10 cm each. Similarly, F is the midpoint of BC, so it's halfway along the right side, also dividing it into two 10 cm segments.Now, quadrilateral BEGF. Let me figure out which points this quadrilateral connects. It should be B, E, G, and F. Wait, but where is point G? The problem doesn't mention point G. Hmm, maybe G is another midpoint or an intersection point? Let me think.Since E and F are midpoints, maybe G is the midpoint of another side? Or perhaps it's the intersection of lines connecting these midpoints. Let me try to sketch this mentally.If I connect E and F, that would be a line from the midpoint of the top side to the midpoint of the right side. But that would form a triangle, not a quadrilateral. Wait, maybe G is another point. Maybe it's the midpoint of CD or DA? Or perhaps it's the intersection of lines from E and F to other sides.Wait, the problem says quadrilateral BEGF, so it must be connecting points B, E, G, and F. So, starting at B, going to E, then to G, then to F, and back to B. So, I need to figure out where G is.Alternatively, maybe G is the intersection point of lines connecting E and F to other midpoints or something. Let me try to think step by step.First, let's assign coordinates to the square to make it easier. Let me place point A at (0, 20), B at (20, 20), C at (20, 0), and D at (0, 0). So, the square is on a coordinate plane with side length 20 cm.Then, point E is the midpoint of AB. Since A is (0, 20) and B is (20, 20), the midpoint E would be at ((0+20)/2, (20+20)/2) = (10, 20).Similarly, point F is the midpoint of BC. B is (20, 20) and C is (20, 0), so midpoint F is at ((20+20)/2, (20+0)/2) = (20, 10).Now, quadrilateral BEGF. So, points B, E, G, F. We know B is (20, 20), E is (10, 20), F is (20, 10). We need to find point G.Wait, maybe G is the midpoint of CD? Let's check. CD goes from (20, 0) to (0, 0), so midpoint would be at (10, 0). But if G is (10, 0), then quadrilateral BEGF would be a four-sided figure connecting (20,20) to (10,20) to (10,0) to (20,10) and back to (20,20). Hmm, that seems a bit odd, but let's see.Alternatively, maybe G is the midpoint of DA? DA goes from (0,0) to (0,20), so midpoint would be (0,10). But connecting (20,20) to (10,20) to (0,10) to (20,10) would form a quadrilateral, but I'm not sure if that's the case.Wait, perhaps G is the intersection of lines from E and F. Let me think. If I draw a line from E (10,20) to some point, and a line from F (20,10) to some point, their intersection might be G.Alternatively, maybe G is the midpoint of another side, but I'm not sure. Let me try to figure this out.Wait, maybe the problem is referring to a standard figure where connecting midpoints forms a smaller square or something. Let me recall that in a square, connecting midpoints of sides can form another square inside.But in this case, we have points E and F as midpoints of AB and BC. If I connect E to F, that's a line from (10,20) to (20,10). If I extend that line, it might intersect with another side or another midpoint.Alternatively, maybe G is the midpoint of CD, which is (10,0). So, if I connect E (10,20) to G (10,0), that's a vertical line. Then, connecting G (10,0) to F (20,10) would be a diagonal line. Hmm, that might form a quadrilateral.Wait, but if G is (10,0), then quadrilateral BEGF would have vertices at (20,20), (10,20), (10,0), and (20,10). Let me plot these points mentally.From (20,20) to (10,20): that's a horizontal line 10 units to the left.From (10,20) to (10,0): that's a vertical line 20 units down.From (10,0) to (20,10): that's a diagonal line going up and to the right.From (20,10) back to (20,20): that's a vertical line 10 units up.Hmm, that seems to form a quadrilateral, but I'm not sure if that's the correct interpretation. Alternatively, maybe G is another point.Wait, perhaps G is the intersection point of lines from E and F to other sides. Let me try that.If I draw a line from E (10,20) to the midpoint of CD, which is (10,0), that's a vertical line. Similarly, if I draw a line from F (20,10) to the midpoint of AD, which is (0,10), that's a horizontal line. Their intersection would be at (10,10). So, point G might be at (10,10).Wait, that makes more sense. So, if G is at (10,10), then quadrilateral BEGF would have vertices at (20,20), (10,20), (10,10), and (20,10). Let me visualize that.From (20,20) to (10,20): horizontal left.From (10,20) to (10,10): vertical down.From (10,10) to (20,10): horizontal right.From (20,10) back to (20,20): vertical up.Yes, that forms a quadrilateral, specifically a trapezoid, since it has two sides parallel (the top and bottom).Wait, but actually, from (10,20) to (10,10) is vertical, and from (20,10) to (20,20) is vertical. So, it's a trapezoid with two vertical sides and two slanting sides? Wait, no, from (10,20) to (10,10) is vertical, and from (10,10) to (20,10) is horizontal. So, actually, it's a rectangle? Wait, no, because from (20,10) to (20,20) is vertical, and from (20,20) to (10,20) is horizontal. So, the figure is a rectangle with vertices at (20,20), (10,20), (10,10), and (20,10). Wait, but that's actually a rectangle, but in reality, it's a square because all sides are equal.Wait, no, let's calculate the lengths. From (20,20) to (10,20) is 10 units. From (10,20) to (10,10) is 10 units. From (10,10) to (20,10) is 10 units. From (20,10) to (20,20) is 10 units. So, all sides are 10 units, so it's a square. But wait, in the coordinate system, it's a square rotated 45 degrees? No, actually, it's axis-aligned.Wait, no, because from (10,10) to (20,10) is horizontal, and from (20,10) to (20,20) is vertical. So, it's a rectangle, but since all sides are equal, it's a square.Wait, but in reality, the distance from (10,10) to (20,10) is 10 units, and from (20,10) to (20,20) is 10 units, so yes, it's a square with side length 10 cm. Therefore, the area would be 10*10=100 cm².But wait, the problem says quadrilateral BEGF. If G is at (10,10), then the area is 100 cm². But let me double-check.Alternatively, maybe G is not at (10,10). Let me think again.If I connect E (10,20) to F (20,10), that's a diagonal line. The equation of that line would be y = -x + 30. Now, if I extend that line, where does it intersect with another side of the square?Wait, but in the square, the sides are at x=0, x=20, y=0, y=20. So, the line y = -x + 30 intersects the square at E (10,20) and F (20,10). If I extend it beyond F, it would go to (30,0), which is outside the square. Similarly, beyond E, it would go to (0,30), which is also outside.So, maybe G is another midpoint. Wait, if I connect E to the midpoint of CD, which is (10,0), that's a vertical line. Similarly, connecting F to the midpoint of AD, which is (0,10), that's a horizontal line. Their intersection is at (10,10), which is the center of the square.So, point G is at (10,10). Therefore, quadrilateral BEGF has vertices at B (20,20), E (10,20), G (10,10), and F (20,10). So, it's a square with side length 10 cm, area 100 cm².Wait, but let me confirm. The coordinates are:B: (20,20)E: (10,20)G: (10,10)F: (20,10)So, plotting these points, it's a square with sides of 10 cm each, so area is 100 cm².But wait, the problem says the side length is 20 cm, so the area of the whole square is 400 cm². If quadrilateral BEGF is 100 cm², that's a quarter of the area. That seems reasonable.Alternatively, maybe I'm overcomplicating it. Let me try another approach.Since E and F are midpoints, the lines from E and F to the opposite sides would intersect at the center of the square, which is (10,10). So, point G is the center.Therefore, quadrilateral BEGF is a square with vertices at (20,20), (10,20), (10,10), and (20,10). So, its area is 10*10=100 cm².Wait, but let me think again. If I divide the square into smaller sections, maybe it's easier.If I connect all midpoints, the square is divided into smaller squares and triangles. Specifically, connecting midpoints of a square divides it into four smaller squares and four triangles.But in this case, we're only connecting two midpoints, E and F, and then to G, which is the center.Alternatively, maybe I can calculate the area using coordinates.Using the coordinates of the four points:B: (20,20)E: (10,20)G: (10,10)F: (20,10)I can use the shoelace formula to calculate the area of quadrilateral BEGF.Shoelace formula is given by:Area = 1/2 |sum over i (x_i y_{i+1} - x_{i+1} y_i)|So, let's list the coordinates in order:(20,20), (10,20), (10,10), (20,10), and back to (20,20).Calculating the terms:First pair: (20,20) and (10,20)x_i y_{i+1} = 20*20 = 400x_{i+1} y_i = 10*20 = 200Difference: 400 - 200 = 200Second pair: (10,20) and (10,10)x_i y_{i+1} = 10*10 = 100x_{i+1} y_i = 10*20 = 200Difference: 100 - 200 = -100Third pair: (10,10) and (20,10)x_i y_{i+1} = 10*10 = 100x_{i+1} y_i = 20*10 = 200Difference: 100 - 200 = -100Fourth pair: (20,10) and (20,20)x_i y_{i+1} = 20*20 = 400x_{i+1} y_i = 20*10 = 200Difference: 400 - 200 = 200Now, sum all the differences: 200 + (-100) + (-100) + 200 = 200 - 100 - 100 + 200 = 200Take the absolute value: |200| = 200Multiply by 1/2: 1/2 * 200 = 100So, the area is 100 cm².Wait, that's consistent with my earlier conclusion. So, quadrilateral BEGF has an area of 100 cm².But let me think again to make sure I didn't make a mistake. The shoelace formula can sometimes give incorrect results if the points are not listed in the correct order or if there's a miscalculation.Let me list the coordinates again:B: (20,20)E: (10,20)G: (10,10)F: (20,10)Back to B: (20,20)Calculating the shoelace terms step by step:First, multiply x_i by y_{i+1}:20*20 = 40010*10 = 10010*10 = 10020*20 = 400Sum: 400 + 100 + 100 + 400 = 1000Next, multiply y_i by x_{i+1}:20*10 = 20020*10 = 20010*20 = 20010*20 = 200Sum: 200 + 200 + 200 + 200 = 800Now, subtract the two sums: 1000 - 800 = 200Take absolute value and multiply by 1/2: 1/2 * 200 = 100Yes, same result. So, the area is indeed 100 cm².Alternatively, I can think of quadrilateral BEGF as a square with side length 10 cm, so area is 10*10=100 cm².Wait, but another way to think about it is that the entire square is 20x20=400 cm². If I divide it into four smaller squares by connecting midpoints, each smaller square would be 10x10=100 cm². So, quadrilateral BEGF is one of those smaller squares, hence 100 cm².Yes, that makes sense. So, the area is 100 cm².But wait, let me make sure that quadrilateral BEGF is indeed one of those smaller squares. If I connect all midpoints, the square is divided into four smaller squares and four triangles. But in this case, quadrilateral BEGF is formed by connecting B, E, G, and F, which are midpoints and the center. So, it's actually one of the smaller squares.Yes, that's correct. So, the area is 100 cm².I think I've confirmed it multiple ways, so I'm confident the answer is 100 cm².

answer：First, I need to determine how many cards Rick gave away in total. He started with 130 cards and decided to keep 15 for himself. So, the total number of cards he gave away is 130 minus 15, which equals 115 cards.Next, Rick gave 13 cards to Miguel. Subtracting this from the total given away, Rick has 115 minus 13, leaving him with 102 cards to distribute further.Rick then saw 8 friends and decided to give each friend 12 cards. To find out how many cards he gave to his friends in total, I multiply the number of friends by the number of cards each friend received: 8 friends multiplied by 12 cards equals 96 cards.Subtracting the 96 cards given to friends from the remaining 102 cards, Rick has 102 minus 96, which leaves him with 6 cards.Finally, Rick wants to give the remaining 6 cards equally to his 2 sisters. To find out how many cards each sister received, I divide the remaining cards by the number of sisters: 6 cards divided by 2 sisters equals 3 cards per sister.

answer：Alright, so I have this inequality to solve: 2x minus x squared is greater than zero. Hmm, okay. Let me write that down to visualize it better: 2x - x² > 0. First, I think I should rearrange this inequality to make it easier to handle. Maybe I can move all terms to one side so that I have a standard quadratic inequality. If I subtract 2x from both sides, I get -x² + 2x > 0. Hmm, that looks a bit messy with the negative coefficient on x squared. Maybe I can factor out a negative sign to make it look nicer. So, factoring out a -1, I get - (x² - 2x) > 0. Wait, but if I factor out a negative, I should remember that it will reverse the inequality sign when I multiply both sides by -1. So, let's see: starting from - (x² - 2x) > 0, if I multiply both sides by -1, the inequality becomes x² - 2x < 0. Okay, that seems better because now I have a positive coefficient on x squared.So, now I have x² - 2x < 0. To solve this quadratic inequality, I should find the roots of the quadratic equation x² - 2x = 0. Factoring that, I can factor out an x: x(x - 2) = 0. So, the roots are x = 0 and x = 2. These roots divide the real number line into three intervals: (-∞, 0), (0, 2), and (2, ∞). To determine where the quadratic expression x² - 2x is less than zero, I can test a value from each interval.Let's start with the interval (-∞, 0). I'll pick x = -1. Plugging into x² - 2x: (-1)² - 2*(-1) = 1 + 2 = 3. That's positive, so the expression is positive in this interval.Next, the interval (0, 2). Let's choose x = 1. Plugging in: 1² - 2*1 = 1 - 2 = -1. That's negative, so the expression is negative here.Lastly, the interval (2, ∞). Let's take x = 3. Plugging in: 3² - 2*3 = 9 - 6 = 3. Positive again.So, the quadratic expression x² - 2x is less than zero only in the interval (0, 2). Therefore, the solution to the inequality x² - 2x < 0 is (0, 2).But wait, let's not forget that we had to multiply both sides by -1 earlier, which flipped the inequality sign. So, going back to the original inequality 2x - x² > 0, the solution is the same as x² - 2x < 0, which is (0, 2). Let me double-check to make sure I didn't make a mistake. If I pick a number in (0, 2), say x = 1, plugging into 2x - x²: 2*1 - 1² = 2 - 1 = 1, which is greater than zero. Good. If I pick x = -1: 2*(-1) - (-1)² = -2 - 1 = -3, which is not greater than zero. And if I pick x = 3: 2*3 - 3² = 6 - 9 = -3, also not greater than zero. So, it seems correct that only (0, 2) satisfies the inequality.Therefore, the solution set is (0, 2), which corresponds to option D.

answer：First, I need to determine the height difference between Vlad and his sister in inches. Both heights are given in feet and inches, so I'll start by converting each person's height entirely into inches.For Vlad:- He is 6 feet, 3 inches tall.- Since there are 12 inches in a foot, I'll multiply the number of feet by 12 to convert the feet to inches.- 6 feet × 12 inches/foot = 72 inches.- Adding the additional 3 inches gives a total of 75 inches.For Vlad's sister:- She is 2 feet, 10 inches tall.- Converting the feet to inches: 2 feet × 12 inches/foot = 24 inches.- Adding the additional 10 inches results in a total of 34 inches.Now, to find out how much taller Vlad is than his sister, I'll subtract his sister's height from Vlad's height:- 75 inches (Vlad) - 34 inches (Sister) = 41 inches.Therefore, Vlad is 41 inches taller than his sister.