Orienteering is a sport that involves navigating your way to a series of checkpoints with the aid of a map and compass. The goal is to find each point on the map as quickly as possible. A typical compass used for orienteering is divided into 360 degree where 0 degrees is north and a bearing is measured clockwise from north.
A.) Gabino and Mia are taking part in an orienteering course. They are instructed to walk 2.3 km on a bearing of 305 degrees to reach point A. There they are supposed to change their bearing to 64 degrees and walk 3.1 km to reach point B. Next they are supposed o return to their original position C. Explain why the interior angle formed at A must be 61 degrees. (The first part of the equation I had to draw the diagram but I don't know how to answer this part) Thanks in advance.
Draw a N-S line through A.
From that line, the angle back to the start is 55°, and the angle to B is 64°
55+64 = 119
That means the interior angle must be 61°, since the entire N-S line is a straight angle.
At point A, sketch in horizontals and verticals
the angle between AC and the x-axis is 35° , (305° - 270°)
So at A by alternate angles and parallel lines the angle between AC and the horizontal is 35°
And AB divides the right angle at A into angles of 64° , (the given angle) and 26°
So at A you have 35° + 26° = 61°
Of course if the rest of the question is to find BC, you would use the Cosine Law
To explain why the interior angle formed at A must be 61 degrees, we need to understand some basic concepts of angles and geometry.
In the given scenario, Gabino and Mia are instructed to walk 2.3 km on a bearing of 305 degrees to reach point A. This means they start from their original position C and walk in a specific direction at an angle of 305 degrees clockwise from north.
Now, let's draw a diagram to visualize the situation. Let's assume that point C is at the center of the diagram, and we draw a line representing the direction of bearing 305 degrees from C. This line will represent the direction in which Gabino and Mia need to walk to reach point A. We also know that they need to walk 2.3 km to reach A.
Next, they are instructed to change their bearing to 64 degrees and walk 3.1 km to reach point B. If we draw another line representing the direction of bearing 64 degrees from A, it will form an angle at point A. Let's call this angle angle X.
To find the measure of angle X, we need to subtract the bearing of 64 degrees from the bearing of 305 degrees. In other words, if we start at north (defined as 0 degrees) and go clockwise to the bearing of 305 degrees, and then continue clockwise to the bearing of 64 degrees, we need to find the difference between these two bearings.
305 degrees - 64 degrees = 241 degrees
Therefore, angle X is equal to 241 degrees.
However, the question asks for the interior angle formed at A, not angle X. The interior angle formed at a given point is always equal to the sum of the adjacent and opposite angles. In this case, angle X is the opposite angle to the interior angle formed at A. Therefore, the interior angle formed at A will be equal to the sum of angle X and the adjacent angle.
Since the sum of the three interior angles of a triangle is always 180 degrees, we can write the equation:
Interior angle at A + angle X + adjacent angle = 180 degrees
Let's assume the adjacent angle to angle X is angle Y.
Interior angle at A + angle X + angle Y = 180 degrees
We know that angle X is 241 degrees, so let's substitute it into the equation:
Interior angle at A + 241 degrees + angle Y = 180 degrees
To simplify the equation, we can rearrange it:
Interior angle at A + angle Y = 180 degrees - 241 degrees
Interior angle at A + angle Y = -61 degrees
Since angles cannot be negative, we can rewrite the equation as:
Interior angle at A + angle Y = 180 degrees + (-61) degrees
Interior angle at A + angle Y = 119 degrees
Now, we know that the sum of the interior angles of a triangle is always 180 degrees. So, we can write another equation using angle Y:
Interior angle at A + angle X + angle Y = 180 degrees
Substituting the values:
Interior angle at A + 241 degrees + angle Y = 180 degrees
Again, we can rearrange the equation:
Interior angle at A + angle Y = 180 degrees - 241 degrees
Interior angle at A + angle Y = -61 degrees
Since angles cannot be negative, we can rewrite the equation as:
Interior angle at A + angle Y = 180 degrees + (-61) degrees
Interior angle at A + angle Y = 119 degrees
Now, we have the equations:
Interior angle at A + angle Y = 119 degrees (Equation 1)
Interior angle at A + angle Y = 119 degrees (Equation 2)
Since both equations are equal to 119 degrees, we can equate them:
Interior angle at A + angle Y = Interior angle at A + angle Y
This tells us that the interior angle at A must be equal to angle Y. Therefore, angle Y is 61 degrees.
In conclusion, the interior angle formed at A must be 61 degrees because it is equal to the adjacent angle to angle X and the sum of the interior angles in a triangle is always 180 degrees.
To explain why the interior angle formed at point A must be 61 degrees, let's break down the problem using basic geometry principles.
1. Draw a diagram: Start by drawing a diagram that represents the given information. Draw a line segment representing the path Gabino and Mia need to walk, starting from their original position C and passing through point A to point B.
2. Measure the bearing angle at point A: Remember that a bearing is measured clockwise from North. In this case, the bearing at point A is 305 degrees. To do this, draw a line from point C in the direction of 305 degrees, which forms an angle with the North line.
3. Analyze the triangle ABC: In the triangle ABC, we have three sides: the distance from point C to point A, the distance from point A to point B, and the distance from point B back to point C (which we can assume is the same as the distance from C to A, based on the instructions).
4. Use the Law of Cosines: The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. In this case, we can use it to find the angle at point A. The formula for the Law of Cosines is:
c^2 = a^2 + b^2 - 2ab * cos(C)
where a, b, and c are the sides of the triangle, and C is the angle opposite side c.
5. Substitute the known values: In our case, side a is the distance from C to A (2.3 km), side b is the distance from A to B (3.1 km), and side c is the distance from B back to C (which we can assume is also 2.3 km). The angle C is the interior angle at point A.
6. Apply the Law of Cosines: By substituting the values into the Law of Cosines equation, we get:
(2.3)^2 = (2.3)^2 + (3.1)^2 - 2(2.3)(3.1) * cos(A)
This simplifies to:
5.29 = 5.29 + 9.61 - 14.26 * cos(A)
7. Solve for cos(A): Rearrange the equation to solve for cosine of angle A:
5.29 - 5.29 - 9.61 = -14.26 * cos(A)
-9.32 = -14.26 * cos(A)
Divide both sides by -14.26:
cos(A) = -9.32 / -14.26
cos(A) = 0.652
8. Find the angle A: Take the inverse cosine (arccos) of 0.652 to find the angle A:
A ≈ 48.85 degrees
9. Calculate the interior angle at point A: Since the interior angles of a triangle add up to 180 degrees, subtract angle A from 180 degrees to find the remaining angle at point A:
180 - 48.85 ≈ 131.15 degrees
Thus, the interior angle formed at point A is approximately 131.15 degrees.