💡Alright, so I've got this geometry problem here, and I need to figure out the equation of line AB and then use one of the given conditions to find the equation of line AC. Let me start by understanding what's given and what I need to find.First, I know that triangle ABC has a vertex at B(-2, 0). There's also an altitude on side AB, and its equation is given as x + 3y - 26 = 0. An altitude of a triangle is a perpendicular line from a vertex to the opposite side. So, this altitude is drawn from vertex C to side AB, right? Wait, no, actually, the altitude is on side AB, which means it's drawn from vertex C to side AB. Hmm, no, actually, the altitude on side AB would be from vertex C to side AB. So, that means point C is somewhere, and the altitude from C to AB is given by x + 3y - 26 = 0.Okay, so I need to find the equation of line AB. Since the altitude is perpendicular to AB, I can use that information to find the slope of AB. The equation of the altitude is x + 3y - 26 = 0. Let me rewrite that in slope-intercept form to find its slope.Starting with x + 3y - 26 = 0, I can rearrange it to 3y = -x + 26, so y = (-1/3)x + 26/3. Therefore, the slope of the altitude is -1/3. Since the altitude is perpendicular to AB, the slope of AB must be the negative reciprocal of -1/3, which is 3. So, the slope of AB is 3.Now that I know the slope of AB and a point on AB, which is B(-2, 0), I can write the equation of AB using the point-slope form. The point-slope form is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.Plugging in the values, I get y - 0 = 3(x - (-2)), which simplifies to y = 3(x + 2). Multiplying that out, I get y = 3x + 6. So, the equation of line AB is y = 3x + 6, or in standard form, 3x - y + 6 = 0.Alright, that takes care of part (1). Now, for part (2), I need to choose one of the given conditions to find the equation of line AC. The two options are:① The equation of the angle bisector of angle A is x + y - 2 = 0.② The equation of the median on side BC is y = 3.I think I'll go with condition ① because it involves an angle bisector, which seems like a more straightforward approach for me. Plus, I remember that angle bisectors have some properties related to the ratio of the sides, which might help in finding point A.So, the angle bisector of angle A is given as x + y - 2 = 0. Since point A is a vertex of the triangle, it must lie on both line AB and the angle bisector. Wait, no, point A is one of the vertices, so it's already on line AB. But the angle bisector is a different line that splits angle A into two equal angles.I need to find the coordinates of point A. Since point A is on line AB, which we've found to be y = 3x + 6, and it's also on the angle bisector x + y - 2 = 0. So, I can set up a system of equations to solve for the coordinates of A.Let me write down the two equations:1. y = 3x + 6 (equation of AB)2. x + y - 2 = 0 (equation of the angle bisector)Substituting equation 1 into equation 2, I get:x + (3x + 6) - 2 = 0Simplify that:x + 3x + 6 - 2 = 0Combine like terms:4x + 4 = 0Subtract 4 from both sides:4x = -4Divide both sides by 4:x = -1Now, plug x = -1 back into equation 1 to find y:y = 3(-1) + 6 = -3 + 6 = 3So, point A is at (-1, 3).Now that I have point A, I need to find the equation of line AC. To do that, I need another point on line AC, which is point C. But I don't know where point C is yet. However, I do know that the altitude from C to AB is given by x + 3y - 26 = 0. Since this altitude passes through point C and is perpendicular to AB, and we already know the slope of AB is 3, so the slope of the altitude is -1/3, which matches the equation given.Wait, but I also know that point C lies on the altitude x + 3y - 26 = 0. So, point C is somewhere on that line. But I don't have enough information yet to find the exact coordinates of C. Maybe I can use the fact that the angle bisector passes through point A and splits angle A into two equal angles.I recall that the angle bisector theorem states that the angle bisector divides the opposite side into segments proportional to the adjacent sides. In other words, the ratio of the lengths of the two segments created by the bisector on the opposite side is equal to the ratio of the lengths of the other two sides of the triangle.But in this case, I'm not sure how to apply that directly because I don't know the lengths of the sides. Maybe there's another way.Alternatively, since I know point A and the angle bisector, perhaps I can find point C by reflecting point B over the angle bisector. Wait, reflecting point B over the angle bisector might give me a point that lies on line AC. I think that's a property of angle bisectors: reflecting one vertex over the angle bisector gives a point on the opposite side.Let me try that. So, if I reflect point B(-2, 0) over the angle bisector x + y - 2 = 0, I should get a point B' that lies on line AC.To find the reflection of a point over a line, I can use the formula for reflection over a line ax + by + c = 0. The formula is:If you have a point (x0, y0), its reflection over the line ax + by + c = 0 is given by:x' = x0 - 2a(ax0 + by0 + c)/(a² + b²)y' = y0 - 2b(ax0 + by0 + c)/(a² + b²)In this case, the angle bisector is x + y - 2 = 0, so a = 1, b = 1, c = -2.Plugging in point B(-2, 0):First, compute ax0 + by0 + c:1*(-2) + 1*0 - 2 = -2 + 0 - 2 = -4Now, compute the denominators:a² + b² = 1 + 1 = 2So,x' = -2 - 2*1*(-4)/2 = -2 - (-8)/2 = -2 + 4 = 2y' = 0 - 2*1*(-4)/2 = 0 - (-8)/2 = 0 + 4 = 4So, the reflection of point B over the angle bisector is point B'(2, 4).Since B' lies on line AC, and we already have point A(-1, 3), we can find the equation of line AC by finding the line passing through points A(-1, 3) and B'(2, 4).First, let's find the slope of AC:m = (4 - 3)/(2 - (-1)) = 1/3So, the slope of AC is 1/3.Using the point-slope form with point A(-1, 3):y - 3 = (1/3)(x - (-1)) => y - 3 = (1/3)(x + 1)Multiplying both sides by 3 to eliminate the fraction:3(y - 3) = x + 1 => 3y - 9 = x + 1 => x - 3y + 10 = 0So, the equation of line AC is x - 3y + 10 = 0.Wait, let me double-check that. If I plug in point A(-1, 3):-1 - 3*3 + 10 = -1 - 9 + 10 = 0, which works.And point B'(2, 4):2 - 3*4 + 10 = 2 - 12 + 10 = 0, which also works.So, that seems correct.Therefore, the equation of line AC is x - 3y + 10 = 0.I think that's it. I used the angle bisector condition to find point A, then reflected point B over the angle bisector to find another point on AC, and then used those two points to find the equation of AC.

💡First, I need to convert the speeds of both trains from kilometers per hour to meters per second to ensure the units are consistent with the given length and time.For the first train:120 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 33.3333 m/sFor the second train:80 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 22.2222 m/sSince the trains are moving in opposite directions, their relative speed is the sum of their individual speeds:33.3333 m/s + 22.2222 m/s = 55.5555 m/sThe total distance covered when the trains cross each other is the sum of their lengths. Let L be the length of the second train:120.00001 m + L = 55.5555 m/s × 9 sCalculating the distance:55.5555 m/s × 9 s = 500.0005 mNow, solve for L:120.00001 m + L = 500.0005 mL = 500.0005 m - 120.00001 mL = 379.99949 mTherefore, the length of the other train is approximately 379.99949 meters.

💡To combine the terms (-2a^{m}b^{4}) and (5a^{n+2}b^{2m+n}) into one term, the exponents of (a) and (b) in both terms must be equal. This gives us two equations to solve:1. For the exponents of (a) to be equal: [ m = n + 2 ]2. For the exponents of (b) to be equal: [ 2m + n = 4 ]From the first equation, we express (m) in terms of (n): [ m = n + 2 ]Substituting (m) into the second equation: [ 2(n + 2) + n = 4 ] [ 2n + 4 + n = 4 ] [ 3n + 4 = 4 ] [ 3n = 0 ] [ n = 0 ]Substituting (n = 0) back into (m = n + 2): [ m = 0 + 2 = 2 ]Thus, (m = 2) and (n = 0). The product (mn) is: [ mn = 2 times 0 = 0 ]Therefore, the value of (mn) is (boxed{0}), which corresponds to choice (boxed{B}).

💡First, I need to determine the total number of pictures Haley took during her vacation. She took 50 pictures at the zoo and 8 pictures at the museum. Adding these together gives a total of 58 pictures.Next, I need to account for the pictures she deleted. Haley deleted 38 pictures from the total she had taken.Finally, to find out how many pictures she still has, I subtract the number of deleted pictures from the total number of pictures taken. So, 58 minus 38 equals 20.Therefore, Haley has 20 pictures remaining from her vacation.

💡Okay, so Joe has 500 toy cars right now. That's a pretty good collection! Now, the problem says he gets 120 more cars. I need to figure out how many cars he'll have in total after getting those additional ones.Alright, let's break this down. He starts with 500 cars. Adding more cars means I need to perform an addition. So, the basic operation here is addition. I'll write that down: 500 plus 120.Hmm, let me make sure I'm adding these correctly. 500 is a round number, and 120 is also a round number, so adding them should be straightforward. Let me visualize it. If I have 500 cars and someone gives me 120 more, I just combine both amounts.Wait, let me double-check. Sometimes when adding numbers, especially larger ones, it's easy to make a mistake. So, 500 plus 100 would be 600, and then adding the remaining 20 gives me 620. That seems right.Is there another way to think about this? Maybe breaking down the numbers. 500 can be thought of as 5 hundreds, and 120 is 1 hundred and 2 tens. So, adding the hundreds first: 5 hundreds plus 1 hundred is 6 hundreds, which is 600. Then, adding the 2 tens gives me 620. Yep, that matches what I got before.Just to be thorough, I'll write it out vertically:``` 500+120----- 620```Adding the units place: 0 + 0 = 0. Tens place: 0 + 2 = 2. Hundreds place: 5 + 1 = 6. So, it all adds up to 620.I think I'm confident that 500 plus 120 equals 620. Joe will have 620 toy cars after getting the additional 120.

💡First, I recognize that ( x ) varies inversely as the square of ( y ). This means the relationship can be expressed as:[x = frac{k}{y^2}]where ( k ) is the constant of variation.Next, I use the given values ( x = 1 ) and ( y = 3 ) to find ( k ):[1 = frac{k}{3^2}][1 = frac{k}{9}][k = 9]Now that I have determined ( k = 9 ), I can find the value of ( x ) when ( y = 9 ):[x = frac{9}{9^2}][x = frac{9}{81}][x = frac{1}{9}]Therefore, the value of ( x ) when ( y = 9 ) is (boxed{frac{1}{9}}).