answer：First, I need to determine the total number of cat stickers June and Bonnie have individually after receiving the additional stickers from their grandparents.For June:- She initially has 76 stickers.- She receives 25 more stickers.- Adding these together: 76 + 25 = 101 stickers.For Bonnie:- She initially has 63 stickers.- She also receives 25 more stickers.- Adding these together: 63 + 25 = 88 stickers.Finally, to find the combined total of stickers, I add June's total to Bonnie's total:101 + 88 = 189 stickers.

answer：First, calculate the time it took Kim to drive to her friend's house.Distance = 80 milesSpeed = 50 mphTime = Distance / SpeedTime = 80 miles / 50 mph = 1.6 hoursNext, calculate the time for each detour on the way back.1. **First detour:**10% longer than the original route means the detour is 1.10 times the original distance.Distance = 1.10 * 80 miles = 88 milesSpeed = 40 mphTime = Distance / SpeedTime = 88 miles / 40 mph = 2.2 hours2. **Second detour:**15% longer than the original route means the detour is 1.15 times the original distance.Distance = 1.15 * 80 miles = 92 milesSpeed = 45 mphTime = Distance / SpeedTime = 92 miles / 45 mph ≈ 2.0444 hours3. **Third detour:**20% longer than the original route means the detour is 1.20 times the original distance.Distance = 1.20 * 80 miles = 96 milesSpeed = 50 mphTime = Distance / SpeedTime = 96 miles / 50 mph = 1.92 hoursNow, add up all the times:- Time to friend's house: 1.6 hours- Time at friend's house: 45 minutes = 0.75 hours- Time for first detour: 2.2 hours- Time for second detour: 2.0444 hours- Time for third detour: 1.92 hoursTotal time away from home = 1.6 + 0.75 + 2.2 + 2.0444 + 1.92 hours ≈ 8.5144 hours

answer：Okay, so I have this problem where I need to find the length of the angle bisector of angle C in triangle ABC, given that angle C is 120 degrees. I need to express this length in terms of the lengths of BC and CA. Hmm, let me try to figure this out step by step.First, I remember that an angle bisector in a triangle divides the angle into two equal parts. So, since angle C is 120 degrees, the bisector will split it into two angles of 60 degrees each. That makes sense because 120 divided by 2 is 60. So, the angle bisector will create two smaller triangles within the original triangle ABC.Let me denote the length of BC as 'a' and the length of CA as 'b'. So, BC = a and CA = b. The angle bisector from point C will meet side AB at some point, let's call it D. So, now we have two smaller triangles: triangle ACD and triangle BCD.I think I can use the Angle Bisector Theorem here. The theorem states that the angle bisector divides the opposite side into segments proportional to the adjacent sides. So, in this case, the ratio of AD to DB should be equal to the ratio of AC to BC. That is, AD/DB = AC/BC = b/a.But wait, I'm not sure if that's directly helpful for finding the length of the angle bisector itself. Maybe I need another approach. I recall there's a formula for the length of an angle bisector in a triangle. Let me try to remember it.I think the formula involves the lengths of the sides and the cosine of the angle. Since we have an angle of 120 degrees, which is pretty large, maybe that will play a role in the formula. Let me try to recall the exact formula.After some thinking, I remember that the length of the angle bisector can be found using the formula:k = (2ab cos (θ/2)) / (a + b)Where 'k' is the length of the angle bisector, 'a' and 'b' are the lengths of the sides adjacent to the angle being bisected, and θ is the measure of that angle.In this case, θ is 120 degrees, so θ/2 is 60 degrees. The cosine of 60 degrees is 0.5, which is 1/2. So, plugging that into the formula, we get:k = (2ab * (1/2)) / (a + b)Simplifying that, the 2 and the 1/2 cancel each other out, so we're left with:k = ab / (a + b)Wait, that seems too straightforward. Let me double-check if I applied the formula correctly.Yes, the formula for the length of the angle bisector is indeed k = (2ab cos (θ/2)) / (a + b). Since θ is 120 degrees, θ/2 is 60 degrees, and cos(60°) is 0.5. So, substituting those values in, the formula simplifies to k = ab / (a + b).But just to be thorough, let me derive this formula to make sure I'm not making a mistake.Let's consider triangle ABC with angle C = 120°, sides BC = a, CA = b, and angle bisector CD = k. The angle bisector divides angle C into two angles of 60° each.Using the Law of Cosines in triangle ABC, we can find the length of side AB. The Law of Cosines states that:AB² = a² + b² - 2ab cos(120°)Since cos(120°) is equal to -0.5, this becomes:AB² = a² + b² - 2ab*(-0.5) = a² + b² + abSo, AB = sqrt(a² + b² + ab)Now, the angle bisector CD divides AB into segments AD and DB such that AD/DB = AC/BC = b/a.Let me denote AD = (b/(a + b)) * AB and DB = (a/(a + b)) * AB.So, AD = (b/(a + b)) * sqrt(a² + b² + ab)and DB = (a/(a + b)) * sqrt(a² + b² + ab)Now, let's consider triangle ACD. In this triangle, we have sides AC = b, CD = k, and AD = (b/(a + b)) * sqrt(a² + b² + ab). The angle at C is 60°, so we can use the Law of Cosines here as well.Applying the Law of Cosines to triangle ACD:AD² = AC² + CD² - 2*AC*CD*cos(60°)Plugging in the known values:[(b/(a + b)) * sqrt(a² + b² + ab)]² = b² + k² - 2*b*k*(0.5)Simplifying the left side:(b²/(a + b)²) * (a² + b² + ab) = b² + k² - b*kLet me compute the left side:(b²/(a + b)²)*(a² + b² + ab) = [b²(a² + b² + ab)] / (a + b)²So, the equation becomes:[b²(a² + b² + ab)] / (a + b)² = b² + k² - b*kLet me multiply both sides by (a + b)² to eliminate the denominator:b²(a² + b² + ab) = (b² + k² - b*k)(a + b)²This looks a bit complicated, but maybe we can simplify it.First, let's expand the right side:(b² + k² - b*k)(a² + 2ab + b²)Multiplying term by term:= b²(a² + 2ab + b²) + k²(a² + 2ab + b²) - b*k(a² + 2ab + b²)= b²a² + 2a b³ + b⁴ + k²a² + 2k²ab + k²b² - b k a² - 2a b² k - b³ kNow, let's write the left side:b²(a² + b² + ab) = a²b² + b⁴ + a b³So, equating left and right sides:a²b² + b⁴ + a b³ = b²a² + 2a b³ + b⁴ + k²a² + 2k²ab + k²b² - b k a² - 2a b² k - b³ kLet me subtract a²b² + b⁴ + a b³ from both sides to bring everything to one side:0 = (b²a² + 2a b³ + b⁴ + k²a² + 2k²ab + k²b² - b k a² - 2a b² k - b³ k) - (a²b² + b⁴ + a b³)Simplify term by term:b²a² - a²b² = 02a b³ - a b³ = a b³b⁴ - b⁴ = 0So, the remaining terms:a b³ + k²a² + 2k²ab + k²b² - b k a² - 2a b² k - b³ k = 0Let me factor out terms where possible:Looking at the terms:a b³ - b³ k = b³(a - k)Similarly, -b k a² + k²a² = a²(k² - b k)And 2k²ab - 2a b² k = 2ab(k² - b k)And k²b² remains as is.So, putting it all together:b³(a - k) + a²(k² - b k) + 2ab(k² - b k) + k²b² = 0Hmm, this seems a bit messy. Maybe there's a better approach.Wait, perhaps instead of using the Law of Cosines in triangle ACD, I should use the formula for the length of the angle bisector directly.I recall that the formula for the length of an angle bisector in a triangle is:k = (2ab cos (θ/2)) / (a + b)As I thought earlier, where θ is the angle at C, which is 120 degrees.So, plugging in θ = 120°, we get θ/2 = 60°, and cos(60°) = 0.5.Therefore, k = (2ab * 0.5) / (a + b) = (ab) / (a + b)So, the length of the angle bisector is ab divided by (a + b).Wait, that seems consistent with what I got earlier. So, maybe I didn't need to go through all that Law of Cosines stuff. But it's good to verify.Alternatively, I can use the formula for the length of the angle bisector which is:k = (2ab / (a + b)) * cos(θ/2)Which, again, gives the same result.Therefore, the length of the angle bisector from angle C is ab / (a + b).I think that's the answer. Let me just make sure I didn't make any mistakes in my reasoning.So, to recap:1. Angle C is 120°, so the bisector divides it into two 60° angles.2. The formula for the angle bisector length is k = (2ab cos(θ/2)) / (a + b).3. Substituting θ = 120°, we get k = (2ab * 0.5) / (a + b) = ab / (a + b).Yes, that seems correct. I think I can confidently say that the length of the angle bisector is ab divided by (a + b).

answer：First, I observe that each number in the sequence (3, 13, 23, ldots, 93) ends with the digit 3. This means that each of these numbers leaves a remainder of 3 when divided by 5.Next, I count the number of terms in the sequence. Starting from 3 and ending at 93, with each subsequent number increasing by 10, there are a total of 10 numbers in the sequence.Since each number is congruent to 3 modulo 5, the product of all these numbers modulo 5 can be expressed as (3^{10} mod 5).To simplify (3^{10} mod 5), I use properties of exponents and modular arithmetic:- (3^2 = 9 equiv 4 mod 5)- (3^4 = (3^2)^2 equiv 4^2 = 16 equiv 1 mod 5)- (3^{10} = (3^4)^2 times 3^2 equiv 1^2 times 4 = 4 mod 5)Therefore, the remainder when the product is divided by 5 is 4.

answer：First, I need to understand what it means for only one diagonal to be drawn from a single vertex in a polygon.In any polygon, a diagonal is a line segment connecting two non-adjacent vertices. So, for a vertex to have only one such connection, there must be exactly one other vertex that is not adjacent to it.I'll start by considering the simplest polygons and count the number of diagonals from one vertex:- In a triangle (3 sides), there are no diagonals because all vertices are connected by sides.- In a quadrilateral (4 sides), each vertex has exactly one diagonal connecting it to the non-adjacent vertex.Since the problem specifies that only one diagonal can be drawn from one vertex, the polygon must have exactly four sides.Therefore, the number of sides of the polygon is 4.

answer：First, I need to estimate the value of (3sqrt{2}). To simplify, I can rewrite (3sqrt{2}) as (sqrt{18}) because (3^2 = 9) and (9 times 2 = 18).Next, I'll identify the perfect squares that (sqrt{18}) falls between. I know that (sqrt{16} = 4) and (sqrt{25} = 5). Since (16 < 18 < 25), it follows that (sqrt{16} < sqrt{18} < sqrt{25}).Therefore, the value of (3sqrt{2}) is between 4 and 5. This means the correct choice is B: Between 4 and 5.