Properties Of Triangles

Let the port be the origin O. Let the positions of the two ships after 2 hours be A and B.
Distance of Ship 1 from the port: .
Distance of Ship 2 from the port: .
Now, we need to find the angle between their paths, .
The direction of Ship 1 is E50 N, which means 50 North of the East direction. The angle with the positive x-axis (East) is .
The direction of Ship 2 is S20 E, which means 20 East of the South direction. The angle measured clockwise from South is 20 . The South direction is at . So, the angle of Ship 2 is from the positive x-axis.
The total angle between their paths is the difference between their angles: .
We now have a triangle OAB with sides , , and the included angle . We need to find the length of the third side, AB.
Using the Law of Cosines:
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The distance is .
To simplify the radical: .
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We can find the circumradius using two main methods.
Method 1: Using the formula , where is the area.
First, calculate the semi-perimeter :
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Next, calculate the area using Heron's formula: .
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Now, use the circumradius formula:
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Method 2: Using the Law of Cosines and the Sine Rule.
First, find the cosine of one of the angles, for example, angle C.
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This means angle .
Now, use the extended Sine Rule: .
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Both methods yield the same result.

Step 1: Understanding the Concept
The problem asks for the sum of the inradius (r) and the circumradius (R) of a right-angled triangle, given the lengths of the two legs.
Step 2: Key Formula or Approach
For a right-angled triangle with legs and hypotenuse : 1. The circumradius is half the length of the hypotenuse: . 2. The inradius can be calculated using the formula . A more direct formula for a right triangle is . 3. First, we need to find the hypotenuse using the Pythagorean theorem: .
Step 3: Detailed Explanation
The triangle is right-angled at B. The sides opposite to vertices A, B, C are respectively. So, the legs are and , and the hypotenuse is . Given: , . 1. Find the hypotenuse b: Using the Pythagorean theorem: (Note: ). 2. Calculate the circumradius R: For a right-angled triangle, the circumcenter is the midpoint of the hypotenuse. 3. Calculate the inradius r: Using the formula for a right-angled triangle: 4. Calculate the required sum r + R: Step 4: Final Answer
The sum of the inradius and the circumradius is .

Step 1: Understanding the Concept
This problem provides two relationships between the sides of a triangle ABC. We need to manipulate these equations, likely using standard triangle formulas like the Cosine Rule or Sine Rule, to determine one of the angles.
Step 2: Key Formula or Approach
The Cosine Rule relates the sides of a triangle to the cosine of one of its angles. For angle C, it is: We will manipulate the given equations to isolate an expression that matches the form of the Cosine Rule.
Step 3: Detailed Explanation
Let's analyze the two given equations. Equation 1: From the Cosine Rule for angle A, we have . Substituting our result: This implies that angle . Equation 2: From the Cosine Rule for angle B, we have . Substituting our result: This implies that angle . We need to find the angle , which is angle C. The sum of angles in a triangle is . Step 4: Final Answer
From the given relations, we found angle and angle . Therefore, angle must be .