Katie hikes 5 miles north, 7 miles east, and then 3 miles north again. What number of miles, measured in a straight line, is katie from her starting point?

Options

A. `sqrt(83)`
B. `10`
C. `sqrt(113)`
D. `13`

Answer

To find the straight-line distance from Katie's starting point to her final position after her hike, we can break down the problem step by step.

### Step 1: Determine the total distance traveled in the north direction.
Katie hikes:
- 5 miles north initially
- 3 miles north again

Total distance north = 5 miles + 3 miles = 8 miles

### Step 2: Determine the distance traveled in the east direction.
Katie hikes:
- 7 miles east

Total distance east = 7 miles

### Step 3: Visualize the problem using a right triangle.
We can visualize Katie's movements on a Cartesian plane:
- The vertical leg of the triangle represents the total distance north (8 miles).
- The horizontal leg of the triangle represents the distance east (7 miles).

### Step 4: Apply the Pythagorean theorem.
To find the straight-line distance (hypotenuse) from the starting point to the final position, we can use the Pythagorean theorem:

\[ c = \sqrt{a^2 + b^2} \]

Where:
- \(c\) is the hypotenuse (the distance we want to find),
- \(a\) is the vertical leg (8 miles),
- \(b\) is the horizontal leg (7 miles).

Substituting the values:

\[ c = \sqrt{(8)^2 + (7)^2} \]

### Step 5: Calculate the squares.
Calculating the squares:

\[ (8)^2 = 64 \]
\[ (7)^2 = 49 \]

### Step 6: Add the squares.
Now, add the squares:

\[ 64 + 49 = 113 \]

### Step 7: Take the square root.
Now take the square root to find the hypotenuse:

\[ c = \sqrt{113} \]

### Step 8: Final calculation.
Calculating \(\sqrt{113}\) gives us approximately:

\[ c \approx 10.63 \text{ miles} \]

### Conclusion:
Katie is approximately 10.63 miles away from her starting point.

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