💡To determine the value of ( k ) such that the points ((1, -2)), ((3, k)), and ((6, 2k - 2)) are collinear, we need to ensure that the slopes between any two pairs of points are equal.1. **Calculate the slope between ((1, -2)) and ((3, k)):** [ m_1 = frac{k - (-2)}{3 - 1} = frac{k + 2}{2} ]2. **Calculate the slope between ((3, k)) and ((6, 2k - 2)):** [ m_2 = frac{(2k - 2) - k}{6 - 3} = frac{k - 2}{3} ]3. **Set the slopes equal to each other for collinearity:** [ frac{k + 2}{2} = frac{k - 2}{3} ] Cross-multiplying gives: [ 3(k + 2) = 2(k - 2) ] [ 3k + 6 = 2k - 4 ] Solving for ( k ): [ k = -10 ]Therefore, the value of ( k ) that makes the points collinear is (-10).

💡Okay, so I have this problem about a parallelogram in the Cartesian coordinate plane. The diagonals intersect at (0,1), and I know the equation of the line containing edge AB is x - 2y - 2 = 0. I need to find the equation of the line containing the opposite edge CD. The options are A) x - 2y - 4 = 0, B) x - 2y + 6 = 0, C) x - 2y - 6 = 0, and D) x - 2y + 4 = 0.First, I remember that in a parallelogram, opposite sides are parallel. So, AB is parallel to CD, and AD is parallel to BC. Since AB is given, CD must have the same slope as AB. The equation of AB is x - 2y - 2 = 0. Let me rewrite this in slope-intercept form (y = mx + b) to find the slope.Starting with x - 2y - 2 = 0, I can rearrange it:-2y = -x + 2Divide both sides by -2:y = (1/2)x - 1So, the slope (m) of AB is 1/2. Therefore, the slope of CD must also be 1/2 because they are parallel.Now, I need to find the equation of CD. Since it's parallel, it will have the same slope, so it will be of the form y = (1/2)x + b, where b is the y-intercept. But I need to find the specific equation, so I need more information.I know that the diagonals of a parallelogram bisect each other. That means the point where the diagonals intersect, which is given as (0,1), is the midpoint of both diagonals. So, if I can find a point on CD, I can use the midpoint formula to find another point, which might help me find the equation.Wait, but I don't have any specific points on CD. Maybe I can use the fact that the diagonals intersect at (0,1). Let me think about the properties of parallelograms and their diagonals.In a parallelogram, the diagonals bisect each other, so if I consider diagonal AC and diagonal BD, they both intersect at (0,1). That means (0,1) is the midpoint of both AC and BD.But I don't have coordinates for points A, B, C, or D. Hmm, maybe I need another approach.Since AB and CD are parallel, and I know the slope of AB is 1/2, CD must also have a slope of 1/2. So, the equation of CD will be similar to AB but with a different y-intercept.Let me denote the equation of CD as x - 2y + c = 0, where c is a constant. Since AB is x - 2y - 2 = 0, CD will be x - 2y + c = 0.Now, I need to find the value of c. To do this, I can use the midpoint property. The midpoint of the diagonals is (0,1). Let's assume that point A is on AB and point C is on CD. Since (0,1) is the midpoint of AC, I can use the midpoint formula.But I don't know the coordinates of A or C. Maybe I can pick a point on AB and find its corresponding point on CD such that (0,1) is the midpoint.Let's choose a point on AB. For simplicity, let me find the y-intercept of AB. When x = 0, from the equation x - 2y - 2 = 0, we get:0 - 2y - 2 = 0 => -2y = 2 => y = -1So, point A is (0, -1). Now, since (0,1) is the midpoint of AC, I can find point C.Midpoint formula: midpoint = ((x1 + x2)/2, (y1 + y2)/2)Here, midpoint is (0,1), and point A is (0, -1). Let point C be (x, y). Then:( (0 + x)/2, (-1 + y)/2 ) = (0,1)So,( x/2, (y - 1)/2 ) = (0,1)This gives two equations:x/2 = 0 => x = 0(y - 1)/2 = 1 => y - 1 = 2 => y = 3So, point C is (0, 3). Now, since point C lies on CD, which has the equation x - 2y + c = 0, I can substitute (0,3) into this equation:0 - 2*3 + c = 0 => -6 + c = 0 => c = 6Therefore, the equation of CD is x - 2y + 6 = 0, which corresponds to option B.Wait, let me double-check. If point C is (0,3), and the equation of CD is x - 2y + 6 = 0, plugging in (0,3):0 - 6 + 6 = 0, which works. Also, since AB and CD are parallel, they should have the same slope, which they do (1/2). So, I think this is correct.Alternatively, I could have considered another point on AB and found its corresponding point on CD, but I think this method is sufficient.

💡First, I need to determine the total area of Kimiko's kitchen floor. The kitchen measures 48 inches by 72 inches, so I'll multiply these two dimensions to find the area.Next, each tile covers an area of 6 square inches. To find out how many tiles are needed to cover the entire kitchen floor, I'll divide the total area of the kitchen by the area of one tile.Finally, the result will give me the number of tiles Kimiko needs to purchase for her kitchen retiling project.

💡Okay, so I need to prove that the area of a quadrilateral with non-perpendicular diagonals is equal to tangent of the angle between the diagonals multiplied by the absolute value of (a² + c² - b² - d²) divided by 4. Hmm, that sounds a bit complicated, but let me try to break it down step by step.First, I remember that the area of a quadrilateral can be found using the formula involving the lengths of the diagonals and the sine of the angle between them. The formula is S = (d₁ * d₂ * sinφ)/2, where d₁ and d₂ are the diagonals and φ is the angle between them. But in this problem, they want the area expressed in terms of the sides a, b, c, d and the tangent of φ instead of sine. So, I need to relate the diagonals and the angle φ to the sides.Maybe I can use the law of cosines in some triangles formed by the diagonals. Let me visualize the quadrilateral ABCD with diagonals AC and BD intersecting at point O. Let’s denote the lengths of the segments as AO = m, BO = n, CO = p, and DO = q. So, the diagonals are AC = m + p and BD = n + q.Now, applying the law of cosines to triangles AOB, BOC, COD, and DOA. For triangle AOB, we have:AB² = AO² + BO² - 2 * AO * BO * cosφa² = m² + n² - 2mn cosφSimilarly, for triangle BOC:BC² = BO² + CO² + 2 * BO * CO * cosφb² = n² + p² + 2np cosφFor triangle COD:CD² = CO² + DO² - 2 * CO * DO * cosφc² = p² + q² - 2pq cosφAnd for triangle DOA:DA² = DO² + AO² + 2 * DO * AO * cosφd² = q² + m² + 2qm cosφOkay, so now I have four equations:1. a² = m² + n² - 2mn cosφ2. b² = n² + p² + 2np cosφ3. c² = p² + q² - 2pq cosφ4. d² = q² + m² + 2qm cosφHmm, maybe I can manipulate these equations to find a relationship between a², b², c², d² and the diagonals.Let me try subtracting equation 2 from equation 1:a² - b² = (m² + n² - 2mn cosφ) - (n² + p² + 2np cosφ)a² - b² = m² - p² - 2mn cosφ - 2np cosφa² - b² = (m² - p²) - 2n(m + p) cosφSimilarly, subtracting equation 4 from equation 3:c² - d² = (p² + q² - 2pq cosφ) - (q² + m² + 2qm cosφ)c² - d² = p² - m² - 2pq cosφ - 2qm cosφc² - d² = (p² - m²) - 2q(p + m) cosφNow, if I add these two results together:(a² - b²) + (c² - d²) = (m² - p² + p² - m²) - 2n(m + p) cosφ - 2q(p + m) cosφSimplifying the left side:a² - b² + c² - d² = 0 - 2n(m + p) cosφ - 2q(m + p) cosφFactor out -2(m + p) cosφ:a² + c² - b² - d² = -2(m + p)(n + q) cosφBut wait, m + p is the length of diagonal AC, which is d₁, and n + q is the length of diagonal BD, which is d₂. So, we have:a² + c² - b² - d² = -2 d₁ d₂ cosφTaking absolute value on both sides:|a² + c² - b² - d²| = 2 d₁ d₂ |cosφ|But in the area formula, we have sinφ. I need to relate sinφ and cosφ. I remember that tanφ = sinφ / cosφ, so sinφ = tanφ * cosφ.From the area formula:S = (d₁ d₂ sinφ)/2 = (d₁ d₂ tanφ cosφ)/2But from above, we have 2 d₁ d₂ cosφ = |a² + c² - b² - d²|So, substituting back:S = (d₁ d₂ tanφ cosφ)/2 = (|a² + c² - b² - d²| tanφ)/4Therefore, the area S is equal to (tanφ * |a² + c² - b² - d²|)/4.Wait, let me double-check my steps. When I subtracted the equations, I got a² + c² - b² - d² = -2 d₁ d₂ cosφ. Taking absolute value gives |a² + c² - b² - d²| = 2 d₁ d₂ |cosφ|. Then, since sinφ = tanφ * cosφ, substituting into the area formula:S = (d₁ d₂ sinφ)/2 = (d₁ d₂ tanφ cosφ)/2But from the equation above, 2 d₁ d₂ cosφ = |a² + c² - b² - d²|, so cosφ = |a² + c² - b² - d²| / (2 d₁ d₂)Substituting back into S:S = (d₁ d₂ tanφ * (|a² + c² - b² - d²| / (2 d₁ d₂)))/2Simplify:S = (tanφ * |a² + c² - b² - d²|) / 4Yes, that seems correct. So, the area is indeed equal to tanφ multiplied by the absolute value of (a² + c² - b² - d²) divided by 4.I think that covers all the steps. I started with the area formula involving diagonals and the sine of the angle, used the law of cosines on the triangles formed by the diagonals, manipulated the equations to express the relationship between the sides and the diagonals, and then substituted back into the area formula to get the desired expression. It all seems to fit together logically.

💡First, I need to determine the total number of dogs in the kennel, which is given as 45.Next, I know that 26 of these dogs have long fur. This means that the number of dogs without long fur is 45 minus 26, which equals 19.Out of these 19 dogs without long fur, 8 are neither long-furred nor brown. Therefore, the remaining dogs without long fur must be brown. This gives me 19 minus 8, which equals 11 short-furred brown dogs.Additionally, there are 19 long-furred dogs that are brown.To find the total number of brown dogs, I add the number of short-furred brown dogs (11) to the number of long-furred brown dogs (19), resulting in 30 brown dogs in total.

💡Okay, so I have this problem about a parallelogram with adjacent sides of lengths 6 and 8. I need to figure out which of the given options cannot be the length of one of its diagonals. The options are A: 3, B: 7, C: 10, and D: 15.First, I remember that in a parallelogram, the diagonals bisect each other. Also, there's a property related to the lengths of the diagonals and the sides. I think it has something to do with the triangle inequality theorem because the diagonals essentially form triangles with the sides of the parallelogram.Let me recall the triangle inequality theorem. It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. So, if I consider one of the diagonals, it should satisfy this condition with the two sides of the parallelogram.Let me denote the sides as a = 6 and b = 8. Let's say the diagonals are d1 and d2. I think the formulas for the diagonals in a parallelogram are related to the sides and the angles between them. The formulas are:d1 = √(a² + b² + 2ab cosθ)d2 = √(a² + b² - 2ab cosθ)Where θ is one of the angles of the parallelogram.Hmm, but maybe I don't need to get into the exact formulas. Instead, I can think about the range of possible lengths for the diagonals. Since the diagonals are related to the sides and the angles, their lengths can vary depending on the angle θ.I remember that the length of a diagonal in a parallelogram must be less than the sum of the two adjacent sides and greater than the absolute difference of the two sides. Wait, is that right? Let me think.Actually, for a triangle, the third side must be less than the sum and greater than the difference of the other two sides. Since the diagonals split the parallelogram into triangles, each diagonal must satisfy the triangle inequality with the two sides.So, for each diagonal, it must be greater than |a - b| and less than a + b.Given that a = 6 and b = 8, let's compute |a - b| and a + b.|6 - 8| = 26 + 8 = 14So, each diagonal must be greater than 2 and less than 14.Looking at the options, A is 3, which is greater than 2, so that's possible. B is 7, which is also between 2 and 14. C is 10, which is still within that range. D is 15, which is greater than 14, so that's not possible.Wait, but I want to make sure I'm not missing something. Is there a case where the diagonal could be equal to 14? If θ is 0 degrees, meaning the parallelogram is degenerate into a straight line, then the diagonal would be 14. But since θ can't be 0 degrees in a proper parallelogram, the diagonal must be less than 14.Similarly, if θ is 180 degrees, the parallelogram is again degenerate, and the diagonal would be |6 - 8| = 2. But again, θ can't be 180 degrees in a proper parallelogram, so the diagonal must be greater than 2.Therefore, the diagonal must be strictly between 2 and 14. So, 15 is outside this range, making it impossible.Let me double-check with the formulas I mentioned earlier. If I plug in θ = 0 degrees, cosθ = 1, so d1 = √(36 + 64 + 2*6*8*1) = √(100 + 96) = √196 = 14. But as I said, this is a degenerate case.For θ = 180 degrees, cosθ = -1, so d2 = √(36 + 64 - 2*6*8*(-1)) = √(100 + 96) = √196 = 14 as well. Wait, that can't be right. If θ is 180 degrees, the parallelogram collapses, and the diagonal should be |6 - 8| = 2, not 14. Maybe I made a mistake in the formula.Let me re-examine the diagonal formulas. I think I might have mixed up the formulas. The correct formulas should be:d1² = a² + b² + 2ab cosθd2² = a² + b² - 2ab cosθSo, if θ = 0 degrees, cosθ = 1, so d1² = 36 + 64 + 96 = 196, so d1 = 14. That's correct.If θ = 180 degrees, cosθ = -1, so d2² = 36 + 64 - (-96) = 36 + 64 + 96 = 196, so d2 = 14. Wait, that still gives 14, but in reality, when θ is 180 degrees, the parallelogram becomes a straight line, and the diagonal should be |6 - 8| = 2. There's a discrepancy here.I think the issue is that when θ = 180 degrees, the parallelogram isn't just collapsing into a line, but the sides are overlapping in the opposite direction. So, perhaps the formula still gives 14 because it's considering the length in the opposite direction. Maybe I need to think differently.Alternatively, perhaps the maximum possible diagonal is 14, and the minimum possible diagonal is 2, but in reality, the diagonals can't actually reach those extremes because the parallelogram can't be completely flat. So, the diagonals must be strictly between 2 and 14.Therefore, 15 is definitely outside that range, so it can't be the length of a diagonal.Just to be thorough, let's check each option:A: 3. Since 3 > 2 and 3 < 14, it's possible.B: 7. Similarly, 7 is within the range.C: 10. Also within the range.D: 15. Outside the range, so impossible.Therefore, the answer should be D: 15.