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Listen\newlineThree friends, Hope, Roger and Catarina, are camping in the woods. They each have their own tent and the tents are set up in a triangle. Hope and Catarina are 8 m8\,\text{m} apart. The angle formed at Catarina is 65∘65^{\circ}. The angle formed at Roger is 42∘42^{\circ}. How far apart are Roger and Catarina?

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Q. Listen\newlineThree friends, Hope, Roger and Catarina, are camping in the woods. They each have their own tent and the tents are set up in a triangle. Hope and Catarina are 8 m8\,\text{m} apart. The angle formed at Catarina is 65∘65^{\circ}. The angle formed at Roger is 42∘42^{\circ}. How far apart are Roger and Catarina?
  1. Law of Sines Explanation: To solve this problem, we will use the Law of Sines, which relates the lengths of sides of a triangle to the sines of its angles. The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. That is, for triangle ABC with sides aa, bb, and cc opposite angles AA, BB, and CC respectively, the following is true: asin⁡(A)=bsin⁡(B)=csin⁡(C)\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}.
  2. Find Third Angle: First, we need to find the third angle of the triangle at Hope. We know that the sum of the angles in any triangle is 180180 degrees. Given that the angle at Catarina is 6565 degrees and the angle at Roger is 4242 degrees, we can find the angle at Hope by subtracting the sum of these two angles from 180180 degrees.\newlineAngle at Hope = 180−(65+42)=180−107=73180 - (65 + 42) = 180 - 107 = 73 degrees.
  3. Calculate Distance Ratio: Now that we have all three angles, we can use the Law of Sines to find the distance between Roger and Catarina. We have the distance between Hope and Catarina (8 m8\,\text{m}) and the angles at Hope (73∘73^\circ) and Catarina (65∘65^\circ). We can set up the ratio as follows:\newlinesin⁡(65∘)8=sin⁡(42∘)RC\frac{\sin(65^\circ)}{8} = \frac{\sin(42^\circ)}{RC}, where RCRC is the distance between Roger and Catarina that we want to find.
  4. Solve for RC: Rearrange the equation to solve for RCRC:RC=8×sin⁡(42)sin⁡(65)RC = \frac{8 \times \sin(42)}{\sin(65)}.
  5. Calculate RC Value: Now we calculate the value of RC using the sine values for the angles: RC≈8×0.6691/0.9063≈8×0.7384≈5.9072RC \approx 8 \times 0.6691 / 0.9063 \approx 8 \times 0.7384 \approx 5.9072 meters.
  6. Round to Two Decimal Places: We round the distance to two decimal places for practical purposes: RC≈5.91RC \approx 5.91 meters.

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