💡First, I need to determine the possible integer lengths for the third side of the triangle, given that the other two sides are 6 and 10 units long.I'll start by applying the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.1. **First Inequality:** [ 6 + 10 > n ] Simplifying this, I get: [ 16 > n ] This means that ( n ) must be less than 16.2. **Second Inequality:** [ 6 + n > 10 ] Simplifying this, I get: [ n > 4 ] This means that ( n ) must be greater than 4.3. **Third Inequality:** [ 10 + n > 6 ] Simplifying this, I get: [ n > -4 ] Since lengths cannot be negative, this inequality doesn't provide any additional information.Combining the relevant inequalities, I have:[4 < n < 16]This means that ( n ) can take integer values from 5 up to 15.Listing these integer values:[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]Counting these values, there are 11 possible integer lengths for the third side.

💡Okay, so I have this problem here: The point with coordinates (10, -14) is the midpoint of the segment with one endpoint at (12, -6). I need to find the sum of the coordinates of the other endpoint. Hmm, let me think about how to approach this.First, I remember that the midpoint of a segment is the average of the coordinates of its endpoints. That is, if I have two endpoints, say (x₁, y₁) and (x₂, y₂), then the midpoint (x_m, y_m) can be found using the formulas:x_m = (x₁ + x₂) / 2y_m = (y₁ + y₂) / 2In this problem, I know the midpoint is (10, -14) and one of the endpoints is (12, -6). I need to find the other endpoint, let's call it (x, y). Once I find x and y, I can just add them together to get the sum.So, let's set up the equations based on the midpoint formulas.For the x-coordinate:10 = (12 + x) / 2And for the y-coordinate:-14 = (-6 + y) / 2Okay, now I need to solve these equations for x and y.Starting with the x-coordinate equation:10 = (12 + x) / 2To solve for x, I can multiply both sides by 2 to get rid of the denominator:10 * 2 = 12 + x20 = 12 + xNow, subtract 12 from both sides:20 - 12 = x8 = xSo, x is 8.Now, moving on to the y-coordinate equation:-14 = (-6 + y) / 2Again, I'll multiply both sides by 2 to eliminate the denominator:-14 * 2 = -6 + y-28 = -6 + yNow, add 6 to both sides to solve for y:-28 + 6 = y-22 = ySo, y is -22.Therefore, the other endpoint is (8, -22). To find the sum of its coordinates, I just add 8 and -22 together:8 + (-22) = -14Wait, that seems straightforward. Let me double-check my calculations to make sure I didn't make any mistakes.For the x-coordinate:10 = (12 + x) / 2Multiply both sides by 2: 20 = 12 + xSubtract 12: x = 8. That looks correct.For the y-coordinate:-14 = (-6 + y) / 2Multiply both sides by 2: -28 = -6 + yAdd 6: y = -22. That also looks correct.Adding them together: 8 + (-22) = -14. Yep, that's right.I think I've got it. The sum of the coordinates of the other endpoint is -14.

💡Alright, so I have this triangle ABC, and I know a few things about it. The side opposite angle A is a, the side opposite angle B is b, and the side opposite angle C is c. They've given me that b is 2, angle A is 60 degrees, and in the first part, a is 3. I need to find sin B. Then, in the second part, if the triangle has two solutions, I need to find the range of values for a.Okay, starting with the first part. I remember the Law of Sines, which says that in any triangle, the ratio of a side to the sine of its opposite angle is constant. So, that would be a/sin A = b/sin B = c/sin C. Since I know a, b, and angle A, I can use this to find sin B.Let me write that down:a / sin A = b / sin BPlugging in the known values:3 / sin 60° = 2 / sin BI know that sin 60° is √3 / 2. So substituting that in:3 / (√3 / 2) = 2 / sin BSimplifying the left side:3 divided by (√3 / 2) is the same as 3 multiplied by 2 / √3, which is 6 / √3. I can rationalize the denominator by multiplying numerator and denominator by √3:6 / √3 = (6√3) / 3 = 2√3So now I have:2√3 = 2 / sin BI can solve for sin B by rearranging:sin B = 2 / (2√3) = 1 / √3Again, rationalizing the denominator:1 / √3 = √3 / 3So sin B is √3 / 3. That seems straightforward.Now, moving on to the second part. They mention that if the triangle has two solutions, I need to find the range of values for a. Hmm, two solutions usually refer to the ambiguous case in the Law of Sines, where given two sides and a non-included angle, there can be two possible triangles. This happens when the given side opposite the given angle is longer than the other given side but shorter than the height of the triangle.Wait, let me think. In this case, we have side b = 2, angle A = 60°, and side a. So, if we fix angle A and side b, varying side a can lead to different triangles. For there to be two solutions, side a must be such that when we draw the triangle, there are two possible positions for point C, making two different triangles.The condition for two solutions is that the length of side a must be greater than the height of the triangle but less than side b. The height can be calculated using the formula: height = b * sin A.So, let's compute the height:height = b * sin A = 2 * sin 60° = 2 * (√3 / 2) = √3So, the height is √3. Therefore, for two solutions, a must be greater than √3 and less than b, which is 2.So, the range of a is √3 < a < 2.Wait, let me verify that. If a is equal to √3, then the triangle would be a right triangle, and there would be only one solution. If a is less than √3, then the side a is too short to reach the base, so no triangle exists. If a is equal to 2, then the triangle would be isoceles with sides a and b equal, so again only one solution. If a is greater than 2, then side a is longer than b, so again only one solution.Therefore, only when a is between √3 and 2 do we have two possible triangles.So, putting it all together, sin B is √3 / 3, and the range for a is (√3, 2).I think that makes sense. I don't see any mistakes in my reasoning. The Law of Sines was applied correctly, and the conditions for two solutions in the ambiguous case were considered properly.

💡Okay, so I have this problem where I need to find vector **b**. The problem says that vectors **a** and **b** add up to another vector, which is [8, -1, -4]. Also, **a** is parallel to [2, 2, 2], and **b** is orthogonal to [2, 2, 2]. Hmm, okay, let me try to figure this out step by step.First, since **a** is parallel to [2, 2, 2], that means **a** is just a scalar multiple of that vector. So I can write **a** as some scalar multiplied by [2, 2, 2]. Let me call that scalar 's'. So, **a** = s*[2, 2, 2] = [2s, 2s, 2s]. That makes sense because if two vectors are parallel, one is just a scaled version of the other.Now, since **a** + **b** = [8, -1, -4], I can write **b** as [8, -1, -4] - **a**. Substituting the expression for **a**, **b** becomes [8 - 2s, -1 - 2s, -4 - 2s]. Okay, so **b** is expressed in terms of 's'.But there's another condition: **b** must be orthogonal to [2, 2, 2]. I remember that two vectors are orthogonal if their dot product is zero. So, I need to take the dot product of **b** and [2, 2, 2] and set it equal to zero.Let me compute that dot product. The dot product of [8 - 2s, -1 - 2s, -4 - 2s] and [2, 2, 2] is:(8 - 2s)*2 + (-1 - 2s)*2 + (-4 - 2s)*2.Let me calculate each term:First term: (8 - 2s)*2 = 16 - 4s.Second term: (-1 - 2s)*2 = -2 - 4s.Third term: (-4 - 2s)*2 = -8 - 4s.Now, adding all these together:16 - 4s - 2 - 4s - 8 - 4s.Combine like terms:16 - 2 - 8 = 6.-4s - 4s - 4s = -12s.So, the equation becomes 6 - 12s = 0.Solving for 's':6 - 12s = 0-12s = -6s = (-6)/(-12) = 1/2.Okay, so s is 1/2. Now, let's substitute s back into the expression for **b**.**b** = [8 - 2*(1/2), -1 - 2*(1/2), -4 - 2*(1/2)].Calculating each component:First component: 8 - 1 = 7.Second component: -1 - 1 = -2.Third component: -4 - 1 = -5.So, **b** is [7, -2, -5].Let me double-check to make sure I didn't make any mistakes. First, check if **a** is indeed parallel to [2, 2, 2]. Since **a** = [2*(1/2), 2*(1/2), 2*(1/2)] = [1, 1, 1], which is indeed a scalar multiple (s=1/2) of [2, 2, 2]. Good.Next, check if **a** + **b** equals [8, -1, -4]. **a** is [1, 1, 1], and **b** is [7, -2, -5]. Adding them together: [1+7, 1+(-2), 1+(-5)] = [8, -1, -4]. Perfect.Lastly, check if **b** is orthogonal to [2, 2, 2]. Compute the dot product: 7*2 + (-2)*2 + (-5)*2 = 14 - 4 - 10 = 0. Yes, that works.So, I think I did everything correctly. **b** is [7, -2, -5].**Final Answer**The vector (mathbf{b}) is boxed{begin{pmatrix} 7 -2 -5 end{pmatrix}}.

💡First, I need to determine which set of three line segments can form a right triangle. To do this, I'll use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.I'll examine each option one by one.**Option A: 1, 1, 1**- Check if (1^2 + 1^2 = 1^2): [1 + 1 = 2 neq 1]- This doesn't satisfy the Pythagorean theorem, so it cannot form a right triangle.**Option B: 2, 3, 4**- Check if (2^2 + 3^2 = 4^2): [4 + 9 = 13 neq 16]- This also doesn't satisfy the theorem, so it can't form a right triangle.**Option C: (sqrt{5}), 3, 4**- Check if (3^2 + (sqrt{5})^2 = 4^2): [9 + 5 = 14 neq 16]- Again, this doesn't satisfy the theorem, so it can't form a right triangle.**Option D: 1, (sqrt{3}), 2**- Check if (1^2 + (sqrt{3})^2 = 2^2): [1 + 3 = 4 = 4]- This satisfies the Pythagorean theorem, meaning it can form a right triangle.After evaluating all options, only option D meets the criteria for forming a right triangle.

💡Okay, so I have this problem here where I'm given two vectors, a and b. Vector a is (2, -3, 5) and vector b is (4, x, y). The question says that these vectors are parallel, and I need to find the values of x and y. The options are A: 6 and -10, B: -6 and 10, C: -6 and -10, D: 6 and 10. Hmm, okay, let me think about how to approach this.First, I remember that if two vectors are parallel, one must be a scalar multiple of the other. That means there exists some scalar, let's call it λ (lambda), such that vector a equals λ times vector b. So, mathematically, that would be:a = λ * bBreaking that down into components, each component of vector a should be equal to λ times the corresponding component of vector b. So, for the first component, 2 should equal λ times 4. For the second component, -3 should equal λ times x. And for the third component, 5 should equal λ times y.Let me write that out:2 = λ * 4 -3 = λ * x 5 = λ * yOkay, so I can solve for λ first using the first equation. If 2 = 4λ, then dividing both sides by 4 gives me λ = 2/4, which simplifies to λ = 1/2. Wait, hold on, that would mean λ is 0.5. But let me double-check that. 4 times 0.5 is 2, which matches the first component. That seems right.Now that I have λ, I can plug that into the second equation to find x. So, -3 = (1/2) * x. To solve for x, I can multiply both sides by 2. That gives me x = -3 * 2 = -6. Okay, so x is -6.Next, using the third equation, 5 = (1/2) * y. Again, I'll solve for y by multiplying both sides by 2. That gives y = 5 * 2 = 10. So, y is 10.Putting it all together, x is -6 and y is 10. Looking back at the options, that's option B: -6 and 10.Wait a second, let me make sure I didn't make a mistake. If λ is 1/2, then vector b should be twice vector a. Let me check:Vector a is (2, -3, 5). If I multiply vector a by 2, I get (4, -6, 10). Comparing that to vector b, which is (4, x, y), that would mean x is -6 and y is 10. Yep, that matches what I found earlier. So, I think I did that correctly.But just to be thorough, let me consider if there's another way to approach this. Maybe using the concept of direction ratios or something else? Hmm, I think the method I used is the standard one for determining if vectors are parallel. They must be scalar multiples of each other.Another thing I could do is check the ratios of the corresponding components. For vectors to be parallel, the ratios of their corresponding components should be equal. So, 2/4 should equal -3/x and also equal 5/y.Calculating 2/4 gives 1/2. So, -3/x should equal 1/2. Solving for x, I get x = -3 * 2 = -6. Similarly, 5/y should equal 1/2, so y = 5 * 2 = 10. Yep, same results. So, that confirms it.I think I'm confident now that x is -6 and y is 10, which is option B. I don't see any mistakes in my calculations, and both methods I used gave me the same answer. So, I think that's the correct choice.**Final Answer**boxed{B}