This problem involves a triangle with the three points: Karsen's starting point, the pond, and the playground. We are tasked with finding the length of the third side of the triangle, where Karsen walks east to return to her starting point. This is a right triangle, and we can use the Pythagorean theorem to find the unknown side.

Steps:

1. The distance from the starting point to the pond is 14 meters south, which we can think of as one leg of the right triangle.
2. The second leg is the horizontal distance (east-west), but since she walks northwest to the playground, we need to break down her northwest movement into components.

When Karsen walks 27 meters northwest, her movement forms an angle of 45 degrees from both the north-south and east-west axes. We can find the horizontal and vertical components of this movement using trigonometry:

• Horizontal component = $27 \cdot \cos(45^\circ) = 27 \cdot \frac{\sqrt{2}}{2} = 19.1 \, \text{meters}$
• Vertical component = $27 \cdot \sin(45^\circ) = 27 \cdot \frac{\sqrt{2}}{2} = 19.1 \, \text{meters}$

Now, the total vertical distance from the pond to the playground is $19.1 - 14 = 5.1$ meters north.

3. Finally, to return to the starting point, Karsen would need to walk east a distance equal to the horizontal component, which is 19.1 meters.

Thus, Karsen will walk approximately 19.1 meters to return to her starting point.

Final answer:

Karsen will walk 19.1 meters to return to her starting point.

Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. What is the Pythagorean theorem, and how is it applied in geometry?
2. How do trigonometric functions like sine and cosine help break down vector components?
3. What are other ways to solve for distances in right triangles?
4. How does one find angles when given two sides of a right triangle?
5. What is the significance of the 45-degree angle in problems involving northwest/southeast directions?

Tip: When solving right triangle problems, always identify which sides are adjacent, opposite, and the hypotenuse relative to the angle in question!