💡Okay, so I have this problem about triangle XYZ with sides XY = 29, XZ = 30, and YZ = 31. I need to find the length of YJ, where J is the incenter of the triangle. The incenter is the point where the angle bisectors meet, right? So, J is the center of the incircle, which touches all three sides of the triangle.First, I remember that the inradius can be found using the formula r = A/s, where A is the area of the triangle and s is the semi-perimeter. So, maybe I should start by calculating the semi-perimeter.The semi-perimeter s is (29 + 30 + 31)/2. Let me compute that: 29 + 30 is 59, plus 31 is 90. So, 90 divided by 2 is 45. So, s = 45.Next, I need to find the area A. I can use Heron's formula for that, which is A = sqrt[s(s - a)(s - b)(s - c)], where a, b, c are the sides of the triangle. Plugging in the values, we get:A = sqrt[45(45 - 29)(45 - 30)(45 - 31)].Calculating each term inside the square root:45 - 29 = 16,45 - 30 = 15,45 - 31 = 14.So, A = sqrt[45 * 16 * 15 * 14].Let me compute that step by step.First, multiply 45 and 16: 45 * 16 = 720.Then, multiply 15 and 14: 15 * 14 = 210.Now, multiply those two results together: 720 * 210. Hmm, that's a big number. Let me see, 720 * 200 = 144,000, and 720 * 10 = 7,200. So, total is 144,000 + 7,200 = 151,200.So, A = sqrt[151,200].Hmm, sqrt(151,200). Let me see if I can simplify that. Let's factor 151,200.151,200 divided by 100 is 1,512. So, sqrt(151,200) = sqrt(1,512 * 100) = 10 * sqrt(1,512).Now, let's factor 1,512. 1,512 divided by 16 is 94.5, which isn't an integer. Let's try dividing by 9: 1,512 / 9 = 168. So, 1,512 = 9 * 168.168 can be factored as 16 * 10.5, but that's not helpful. Wait, 168 is 16 * 10.5, but 10.5 isn't an integer. Maybe another approach.Wait, 1,512 divided by 144 is 10.5, which again isn't helpful. Maybe I should factor it into primes.1,512: Let's divide by 2: 1,512 / 2 = 756.756 / 2 = 378.378 / 2 = 189.189 / 3 = 63.63 / 3 = 21.21 / 3 = 7.So, the prime factors are 2^3 * 3^3 * 7.So, sqrt(1,512) = sqrt(2^3 * 3^3 * 7) = 2^(1.5) * 3^(1.5) * sqrt(7) = 2 * sqrt(2) * 3 * sqrt(3) * sqrt(7) = 6 * sqrt(42).Wait, let me check that again.sqrt(2^3 * 3^3 * 7) = 2^(1.5) * 3^(1.5) * sqrt(7) = 2*sqrt(2) * 3*sqrt(3) * sqrt(7) = 6 * sqrt(2*3*7) = 6 * sqrt(42).Yes, that's correct.So, sqrt(1,512) = 6 * sqrt(42).Therefore, A = 10 * 6 * sqrt(42) = 60 * sqrt(42).So, the area A is 60 * sqrt(42).Now, the inradius r = A / s = (60 * sqrt(42)) / 45.Simplify that: 60 / 45 = 4/3.So, r = (4/3) * sqrt(42).Hmm, okay. So, the inradius is (4/3) * sqrt(42). But wait, I need to find YJ, which is the distance from vertex Y to the incenter J.I remember that the distance from a vertex to the incenter can be found using the formula:d = sqrt[r^2 + (s - a)^2],where a is the side opposite the vertex. In this case, YJ is the distance from Y to J, so a would be the side opposite Y, which is XZ = 30.Wait, let me confirm. In triangle XYZ, vertex Y is opposite side XZ, which is 30. So, yes, a = 30.So, s - a = 45 - 30 = 15.So, YJ = sqrt[r^2 + (15)^2].We already have r = (4/3) * sqrt(42), so r^2 = (16/9) * 42 = (16 * 42) / 9.Compute 16 * 42: 16*40=640, 16*2=32, so total is 672.So, r^2 = 672 / 9 = 224 / 3.So, YJ = sqrt[(224 / 3) + 225].Wait, 15^2 is 225, so we have sqrt[(224 / 3) + 225].Convert 225 to thirds: 225 = 675 / 3.So, total inside the sqrt is (224 + 675) / 3 = 899 / 3.So, YJ = sqrt(899 / 3).Hmm, that's approximately sqrt(299.666...), which is roughly 17.31. But none of the answer choices are 17.31. The options are 15, 16, 17, 18, 19.Wait, maybe I made a mistake in the formula. Let me double-check.I used the formula d = sqrt[r^2 + (s - a)^2]. Is that correct?Wait, I think I might have confused it with another formula. Let me recall.The distance from the incenter to a vertex can be found using the formula:d = sqrt[r^2 + (s - a)^2],where s is the semi-perimeter and a is the side opposite the vertex.But let me verify this formula. Alternatively, I think the formula is:d = (2 * sqrt[s * (s - a) * (s - b) * (s - c)]) / (a + b + c),but I'm not sure.Wait, maybe it's better to use coordinates to find the exact distance.Let me try another approach. Let's place the triangle in a coordinate system.Let me place point Y at the origin (0, 0), point Z at (31, 0), since YZ = 31. Then, point X is somewhere in the plane. We know XY = 29 and XZ = 30.So, coordinates of X satisfy the distances from Y(0,0) and Z(31,0):Distance from Y: sqrt[(x)^2 + (y)^2] = 29,Distance from Z: sqrt[(x - 31)^2 + (y)^2] = 30.So, we have two equations:x^2 + y^2 = 29^2 = 841,(x - 31)^2 + y^2 = 30^2 = 900.Subtract the first equation from the second:(x - 31)^2 + y^2 - x^2 - y^2 = 900 - 841,Expand (x - 31)^2: x^2 - 62x + 961,So, x^2 - 62x + 961 + y^2 - x^2 - y^2 = 59,Simplify: -62x + 961 = 59,So, -62x = 59 - 961 = -902,Thus, x = (-902)/(-62) = 902 / 62.Simplify 902 / 62: 62*14 = 868, 902 - 868 = 34, so 14 + 34/62 = 14 + 17/31 ≈ 14.548.So, x = 14 + 17/31.Now, plug x back into the first equation to find y:x^2 + y^2 = 841,So, y^2 = 841 - x^2.Compute x^2:x = 14 + 17/31 = (14*31 + 17)/31 = (434 + 17)/31 = 451/31.So, x^2 = (451)^2 / (31)^2.Compute 451^2: 450^2 = 202,500, plus 2*450*1 + 1^2 = 202,500 + 900 + 1 = 203,401.So, x^2 = 203,401 / 961.Thus, y^2 = 841 - (203,401 / 961).Convert 841 to over 961: 841 = 841 * 961 / 961 = 808,201 / 961.So, y^2 = (808,201 - 203,401) / 961 = 604,800 / 961.Simplify 604,800 / 961: Let's see if 961 divides into 604,800.961 * 629 = 961*(600 + 29) = 961*600 + 961*29.961*600 = 576,600,961*29: 961*30 = 28,830, minus 961 = 28,830 - 961 = 27,869.So, total is 576,600 + 27,869 = 604,469.But 604,800 - 604,469 = 331.So, 604,800 = 961*629 + 331.So, y^2 = (961*629 + 331)/961 = 629 + 331/961.So, y^2 = 629 + 331/961 ≈ 629.344.Thus, y ≈ sqrt(629.344) ≈ 25.1.So, coordinates of X are approximately (14.548, 25.1).Now, the incenter J has coordinates given by the formula:J = (a*A_x + b*B_x + c*C_x)/ (a + b + c), similarly for y-coordinates,where a, b, c are the lengths of the sides opposite to vertices A, B, C.Wait, in our case, the triangle is XYZ, with sides XY = 29, XZ = 30, YZ = 31.So, the sides opposite to X, Y, Z are YZ = 31, XZ = 30, XY = 29, respectively.So, the coordinates of J are:J_x = (a*A_x + b*B_x + c*C_x) / (a + b + c),where a = YZ = 31, opposite X,b = XZ = 30, opposite Y,c = XY = 29, opposite Z.Wait, actually, I think the formula is weighted by the lengths of the sides.Wait, let me recall: the incenter coordinates are given by ( (a*x_A + b*x_B + c*x_C)/ (a + b + c), (a*y_A + b*y_B + c*y_C)/ (a + b + c) ).But in our case, the vertices are Y(0,0), Z(31,0), and X(14.548,25.1).So, the sides opposite these vertices are:Opposite Y: XZ = 30,Opposite Z: XY = 29,Opposite X: YZ = 31.So, a = 30, b = 29, c = 31.Wait, no, actually, the formula is:If the triangle has vertices A, B, C, and sides opposite to these vertices are a, b, c respectively, then the incenter is ( (a*A_x + b*B_x + c*C_x)/ (a + b + c), same for y).So, in our case:Vertex Y is opposite side XZ = 30,Vertex Z is opposite side XY = 29,Vertex X is opposite side YZ = 31.So, a = 30, b = 29, c = 31.Thus, incenter J has coordinates:J_x = (30*Y_x + 29*Z_x + 31*X_x) / (30 + 29 + 31),Similarly, J_y = (30*Y_y + 29*Z_y + 31*X_y) / (30 + 29 + 31).Given that Y is (0,0), Z is (31,0), and X is approximately (14.548,25.1).So, compute J_x:= (30*0 + 29*31 + 31*14.548) / (90).Compute each term:30*0 = 0,29*31: 29*30=870, plus 29=899,31*14.548: Let's compute 14.548 * 30 = 436.44, plus 14.548 = 450.988.So, total numerator for J_x: 0 + 899 + 450.988 = 1349.988.So, J_x = 1349.988 / 90 ≈ 14.9999 ≈ 15.Similarly, J_y:= (30*0 + 29*0 + 31*25.1) / 90.Compute:30*0 = 0,29*0 = 0,31*25.1 ≈ 31*25 + 31*0.1 = 775 + 3.1 = 778.1.So, numerator ≈ 778.1,Thus, J_y ≈ 778.1 / 90 ≈ 8.6456.So, coordinates of J are approximately (15, 8.6456).Now, we need to find the distance YJ, which is the distance from Y(0,0) to J(15,8.6456).Using the distance formula:YJ = sqrt[(15 - 0)^2 + (8.6456 - 0)^2] = sqrt[225 + 74.747] ≈ sqrt[300] ≈ 17.32.Wait, but the answer choices are 15,16,17,18,19. So, 17.32 is closest to 17. So, is the answer 17? But earlier, when I tried using the formula, I got sqrt(899/3) ≈ sqrt(299.666) ≈ 17.31, which also rounds to 17.But wait, the options include 17, which is C. But in the initial thought process, the user concluded it was 15, which is A. So, which one is correct?Wait, maybe I made a mistake in the formula earlier. Let me check again.I used d = sqrt[r^2 + (s - a)^2], where a is the side opposite the vertex. So, for YJ, a = XZ = 30, so s - a = 15.Then, r = (4/3)*sqrt(42) ≈ (4/3)*6.4807 ≈ 8.6409.So, r^2 ≈ 74.747,(s - a)^2 = 225,So, d = sqrt(74.747 + 225) = sqrt(300) ≈ 17.32.So, that's consistent with the coordinate method.But in the initial thought process, the user concluded YJ = 15, which seems incorrect because 15 is the length of (s - a), which is 15, but that's just one component of the distance.So, perhaps the initial thought process was wrong, and the correct answer is 17.Wait, but let me think again. Maybe there's another formula.I recall that the distance from the incenter to a vertex can also be expressed as:d = (2 * sqrt[s * (s - a) * (s - b) * (s - c)]) / (a + b + c),Wait, no, that formula is for the length of the angle bisector, not the distance from the incenter.Wait, actually, the formula for the distance from the incenter to a vertex is:d = sqrt[r^2 + (s - a)^2],which is what I used earlier.So, that gives us approximately 17.32, which is about 17.But let me check another way.Alternatively, since we have the coordinates of J as approximately (15, 8.6456), and Y is at (0,0), then YJ is sqrt(15^2 + 8.6456^2) ≈ sqrt(225 + 74.747) ≈ sqrt(300) ≈ 17.32.So, that's consistent.But wait, the answer choices are integers, so 17.32 is closest to 17, which is option C.But in the initial thought process, the user concluded it was 15, which is option A. So, perhaps the user made a mistake.Wait, let me see. The user wrote:"Since J is the incenter, YJ is the bisector of YZ, hence is 15."But that doesn't make sense because YJ is not the bisector of YZ; rather, the angle bisector of angle Y would meet YZ at some point, but YJ is the distance from Y to the incenter, not the length of the bisector segment on YZ.So, the user incorrectly concluded that YJ is 15, which is actually the length of YQ, where Q is the point where the incircle touches YZ.So, YQ = s - a = 15, but YJ is longer than that.Therefore, the correct answer should be approximately 17.32, which is closest to 17, so option C.But wait, let me check if there's an exact value.We had YJ = sqrt[(4/3 * sqrt(42))^2 + 15^2] = sqrt[(16/9 * 42) + 225] = sqrt[(672/9) + 225] = sqrt[(224/3) + 225] = sqrt[(224 + 675)/3] = sqrt[899/3].Now, 899 divided by 3 is approximately 299.666..., whose square root is approximately 17.31.But 899 is 29*31, right? 29*31=899.So, sqrt(899/3) = sqrt(29*31/3).Hmm, that doesn't simplify nicely. So, it's an irrational number approximately 17.31.But the answer choices are integers, so 17 is the closest.Wait, but let me check if I made a mistake in the coordinate method.I placed Y at (0,0), Z at (31,0), and found X at approximately (14.548,25.1). Then, the incenter J was at approximately (15,8.6456). So, YJ is sqrt(15^2 + 8.6456^2) ≈ sqrt(225 + 74.747) ≈ sqrt(300) ≈ 17.32.But wait, 15^2 + (something)^2 = 300, so the something is sqrt(300 - 225) = sqrt(75) ≈ 8.66, which matches the y-coordinate of J.So, that seems consistent.Alternatively, maybe there's a formula that gives YJ directly.I recall that in a triangle, the distance from a vertex to the incenter can be found using the formula:d = (2 * area) / (a + b + c) * (1 + cos(A/2)),where A is the angle at vertex Y.But that seems complicated.Alternatively, using trigonometric identities.Wait, maybe using the formula:d = (r) / sin(A/2),where A is the angle at Y.But to use that, I need to find angle Y.Alternatively, using the formula:d^2 = r^2 + (s - a)^2,which is what I used earlier.So, given that, and since we have r = (4/3)*sqrt(42) and s - a = 15,then YJ = sqrt[(16/9 * 42) + 225] = sqrt[(672/9) + 225] = sqrt[(224/3) + 225] = sqrt[(224 + 675)/3] = sqrt[899/3] ≈ 17.31.So, the exact value is sqrt(899/3), which is approximately 17.31, so the closest integer is 17.Therefore, the answer should be C) 17.But wait, let me check if there's a way to get an exact integer value.Wait, 899/3 is approximately 299.666..., which is not a perfect square. So, it's irrational.But the answer choices are integers, so 17 is the closest.Alternatively, maybe I made a mistake in calculating the inradius.Wait, let's recalculate the inradius.We had area A = 60*sqrt(42),s = 45,so r = A/s = (60*sqrt(42))/45 = (4/3)*sqrt(42).Yes, that's correct.So, r = (4/3)*sqrt(42),r^2 = (16/9)*42 = (16*42)/9 = (672)/9 = 74.666...So, YJ = sqrt(74.666... + 225) = sqrt(300) ≈ 17.32.So, yes, 17 is the closest integer.Therefore, the answer is C) 17.But wait, in the initial thought process, the user concluded it was 15, which is incorrect. So, the correct answer is 17.

💡First, I need to determine the condition for the lines ( L_1 ) and ( L_2 ) to be parallel. Two lines are parallel if their slopes are equal.I'll start by finding the slope of each line.For ( L_1: ax + 3y + 1 = 0 ), I'll solve for ( y ):[3y = -ax - 1 y = -frac{a}{3}x - frac{1}{3}]So, the slope of ( L_1 ) is ( -frac{a}{3} ).For ( L_2: 2x + (a + 1)y + 1 = 0 ), I'll solve for ( y ):[(a + 1)y = -2x - 1 y = -frac{2}{a + 1}x - frac{1}{a + 1}]So, the slope of ( L_2 ) is ( -frac{2}{a + 1} ).To ensure the lines are parallel, their slopes must be equal:[-frac{a}{3} = -frac{2}{a + 1}]Simplifying this equation:[frac{a}{3} = frac{2}{a + 1} a(a + 1) = 6 a^2 + a - 6 = 0]Now, I'll solve the quadratic equation:[a^2 + a - 6 = 0]Using the quadratic formula:[a = frac{-1 pm sqrt{1 + 24}}{2} = frac{-1 pm 5}{2}]This gives two solutions:[a = frac{-1 + 5}{2} = 2 quad text{and} quad a = frac{-1 - 5}{2} = -3]Therefore, the possible values of ( a ) are 2 and -3.

💡First, I recognize that any integer leaving a remainder of 4 when divided by 6 can be expressed in the form (6n + 4), where (n) is an integer.To find the largest such integer less than 100, I set up the inequality:[6n + 4 < 100]Subtracting 4 from both sides gives:[6n < 96]Dividing both sides by 6 yields:[n < 16]The largest integer value for (n) that satisfies this inequality is 15. Substituting (n = 15) into the expression (6n + 4) results in:[6 times 15 + 4 = 90 + 4 = 94]Therefore, the largest integer less than 100 that leaves a remainder of 4 when divided by 6 is 94.

💡Okay, so I have this problem here with two sets, M and N, and I need to find their intersection. Let me try to understand what each set represents.First, set M is defined as {x ∈ ℝ | y = log x}. Hmm, okay. So, M is the set of all real numbers x for which y is equal to the logarithm of x. Wait, logarithm functions have specific domains, right? The logarithm function, log x, is only defined when x is positive. So, that means x has to be greater than zero. Therefore, set M is all positive real numbers. In interval notation, that would be (0, +∞). Got that.Now, moving on to set N, which is {y ∈ ℝ | y = x² + 1}. Alright, so N is the set of all real numbers y such that y is equal to x squared plus one. Let me think about the range of this function. Since x squared is always non-negative (because squaring any real number gives a positive result or zero), the smallest value x² can take is zero. So, when x is zero, y becomes 0 + 1, which is 1. As x increases or decreases from zero, x² increases, making y larger and larger. Therefore, the smallest value y can take is 1, and it goes up to infinity. So, set N is all real numbers y such that y is greater than or equal to 1. In interval notation, that's [1, +∞).Now, I need to find the intersection of M and N, which is M ∩ N. The intersection of two sets consists of all elements that are common to both sets. So, I need to find the overlap between M, which is (0, +∞), and N, which is [1, +∞).Let me visualize this. M starts just above zero and goes to infinity, while N starts at 1 and goes to infinity. The overlap between these two sets would be where both conditions are satisfied. So, starting from 1, since that's where N begins, and going to infinity, because that's where both sets continue. Therefore, the intersection should be [1, +∞).Wait, let me double-check. M is all positive real numbers, so it includes numbers like 0.5, 1, 2, etc. N includes numbers starting from 1 upwards, so 1, 2, 3, etc. The common elements between them are the numbers that are in both sets. So, numbers greater than or equal to 1 are in both M and N. Numbers less than 1 but greater than 0 are only in M, not in N. So, the intersection is indeed [1, +∞).Looking back at the options given:A: (0, +∞) – This is just set M, not the intersection.B: [1, +∞) – This seems to be the correct intersection.C: (-∞, +∞) – This would mean all real numbers, which isn't the case since both sets have restrictions.D: (0,1] – This is the part of M that doesn't overlap with N, so it's not the intersection.Therefore, the correct answer should be option B.**Final Answer**boxed{B}

💡Okay, so I have this quadratic equation: ( ax^2 + bx + c = 0 ), and I know that ( a ) is not zero, which makes sense because if ( a ) were zero, it wouldn't be a quadratic equation anymore. The problem states that ( a + b + c = 0 ) and ( a - b + c = 0 ). I need to find the roots of this equation based on these conditions.First, let me write down the given equations:1. ( a + b + c = 0 )2. ( a - b + c = 0 )Hmm, these look similar but with a sign change on the ( b ) term. Maybe I can subtract or add these equations to eliminate one of the variables. Let me try subtracting the second equation from the first one.Subtracting equation 2 from equation 1:( (a + b + c) - (a - b + c) = 0 - 0 )Simplify:( a + b + c - a + b - c = 0 )Simplify further:( 2b = 0 )So, ( 2b = 0 ) implies that ( b = 0 ). Okay, so the coefficient ( b ) is zero. That simplifies the quadratic equation a bit.Now, let's plug ( b = 0 ) back into one of the original equations to find a relationship between ( a ) and ( c ). Let's use equation 1:( a + 0 + c = 0 )So, ( a + c = 0 ), which means ( c = -a ).Now, let's substitute ( b = 0 ) and ( c = -a ) into the quadratic equation:( ax^2 + 0x + (-a) = 0 )Simplify:( ax^2 - a = 0 )Factor out ( a ):( a(x^2 - 1) = 0 )Since ( a neq 0 ), we can divide both sides by ( a ):( x^2 - 1 = 0 )Now, solving for ( x ):( x^2 = 1 )Taking square roots:( x = pm 1 )So, the roots are ( x = 1 ) and ( x = -1 ).Let me double-check to make sure I didn't make any mistakes. Starting from the given conditions:1. ( a + b + c = 0 )2. ( a - b + c = 0 )Subtracting the second from the first gives ( 2b = 0 ), so ( b = 0 ). Then substituting back, ( a + c = 0 ) gives ( c = -a ). Plugging into the quadratic equation, we get ( ax^2 - a = 0 ), which factors to ( a(x^2 - 1) = 0 ). Since ( a neq 0 ), ( x^2 - 1 = 0 ) leads to ( x = 1 ) and ( x = -1 ). That seems correct.I think the answer is C: 1, -1.

💡Okay, so I have this problem about triangle ABC. It says that the lengths of vectors AB and AC are both 4, and the dot product of AB and AC is 8. I need to figure out what shape this triangle is. Hmm, let me think.First, I remember that in a triangle, if two sides are equal, it's called an isosceles triangle. So since AB and AC are both 4 units long, triangle ABC is at least isosceles. But I need to determine if it's something more specific, like equilateral or maybe a right triangle.The problem also gives me the dot product of AB and AC, which is 8. I recall that the dot product of two vectors can be calculated using the formula:[overrightarrow{AB} cdot overrightarrow{AC} = |overrightarrow{AB}| |overrightarrow{AC}| cos theta]where θ is the angle between vectors AB and AC. In this case, that angle is angle A of the triangle.Given that both |AB| and |AC| are 4, I can plug those values into the formula:[8 = 4 times 4 times cos theta]Simplifying that, I get:[8 = 16 cos theta]Dividing both sides by 16:[cos theta = frac{8}{16} = frac{1}{2}]So, θ is the angle whose cosine is 1/2. I remember that cos(60°) = 1/2, which is π/3 radians. So angle A is 60 degrees.Now, in an isosceles triangle, if the angle between the two equal sides is 60 degrees, then the triangle is actually equilateral. Because in an equilateral triangle, all sides are equal and all angles are 60 degrees. Since AB and AC are equal and the angle between them is 60 degrees, that means the third side BC must also be equal to AB and AC.Let me double-check that. If angle A is 60 degrees and sides AB and AC are equal, then sides BC can be found using the Law of Cosines:[BC^2 = AB^2 + AC^2 - 2 times AB times AC times cos A]Plugging in the values:[BC^2 = 4^2 + 4^2 - 2 times 4 times 4 times cos 60°]Calculating each term:[BC^2 = 16 + 16 - 32 times frac{1}{2}][BC^2 = 32 - 16][BC^2 = 16][BC = 4]So BC is also 4, which means all sides are equal. Therefore, triangle ABC is equilateral.Wait, just to make sure I didn't make a mistake, let me think again. If two sides are equal and the angle between them is 60 degrees, does that necessarily make the third side equal? Yes, because the Law of Cosines shows that BC comes out equal to AB and AC. So, yes, it's equilateral.I don't think I made any calculation errors. The dot product gave me the angle, which turned out to be 60 degrees, and with the two sides equal, that forces the third side to be equal too. So, the triangle must be equilateral.**Final Answer**The triangle is an equilateral triangle, so the shape is boxed{text{Equilateral triangle}}.