To determine if the biker's estimate of the distance from their starting point is correct, we can use the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the biker travels 5 kilometers east and 12 kilometers north, which forms a right triangle:
- One side (eastward) = 5 km
- Another side (northward) = 12 km
- Hypotenuse (direct distance from the starting point) = d km
According to the Pythagorean Theorem:
\[ d^2 = 5^2 + 12^2 \]
Calculating this gives:
\[ d^2 = 25 + 144 = 169 \] \[ d = \sqrt{169} = 13 \text{ km} \]
The biker estimated the distance to be 14 kilometers, but the actual distance calculated using the Pythagorean Theorem is 13 kilometers. Thus, the biker is incorrect in their estimate.
Now, regarding the justification:
- The Pythagorean Theorem is used to find the relationship between the sides of a right triangle.
- The converse of the Pythagorean Theorem is used to determine if a triangle with given side lengths is a right triangle.
Since we used the Pythagorean Theorem to establish the actual distance, the appropriate response would be:
The biker is incorrect, according to the Pythagorean Theorem.