A ship in calm seas steamed 13 km in one direction, turned and steamed 13 km in another
direction, and then returned 12 km back to its original position. The captain then plotted
the ship’s course on a nautical chart. She asked her first officer to look at the chart and
describe the ship’s path. Did the first officer describe it as an acute, obtuse, or right
triangle? Then the second officer said she could further identify whether the path was
scalene, isosceles, or equilateral. What did she determine?
I'm not sure, but I think it might be acute; isosceles?
Looks like you end up with a triangle with sides
13, 13, and 12.
mmmhhh, two of the sides are equal, and the third is almost equal to the other two.
So it is "almost" an equilateral triangle, and the angles should be "near" 60°
How can you NOT be sure about it being acute and isosceles ?
To determine the shape of the ship's path, we need to understand the distances and angles involved.
First, let's draw the ship's path on a coordinate system. We will assume the ship starts at the origin (0,0). The ship steams 13 km in one direction, so it moves to the point (13,0).
Next, the ship turns and steams 13 km in another direction. We don't know the angle of the turn, but since the ship is backtracking, we can infer that it turned around 180 degrees. This means the ship reaches the point (-13,0).
Finally, the ship returns 12 km back to its original position, which means it moves from (-13,0) to (0,0).
Now let's analyze the triangle formed by these three points: (13,0), (-13,0), and (0,0).
We can see that two sides of the triangle have lengths 13 km, while the third side has length 12 km. Therefore, the triangle is not equilateral.
To determine if the triangle is acute, obtuse, or right-angled, we need to examine the angles. In this case, the triangle has a 90-degree angle at the origin (0,0), since it moves from (-13,0) to (0,0), forming a right angle.
Based on this analysis, the first officer correctly described the shape of the ship's path as an "acute triangle."
Now, the second officer can determine whether the triangle is scalene, isosceles, or equilateral. Since two sides of the triangle have lengths 13 km, it is not scalene. However, since the triangle has a right angle, it is not isosceles either. Therefore, the second officer can determine that the ship's path is an "acute and scalene triangle."