A person walked completely around the edge of a park beginning at the midpoint of one edge and making the...
GMAT Data Sufficiency : (DS) Questions
A person walked completely around the edge of a park beginning at the midpoint of one edge and making the minimum number of turns, each with the minimum number of degrees necessary, as shown in the figure above. What is the sum of the degrees of all the turns that the person made?
- One of the turns is \(80°\).
- The number of sides of the park is \(4\), all of the sides are straight, and each interior angle is less than \(180°\).
Understanding the Question
We need to find the sum of all the turn angles made when walking completely around a park's edge.
Given Information
- Person walks completely around the park's edge
- Starts at the midpoint of one edge
- Makes minimum number of turns with minimum degrees
- Park has some polygon shape
What We Need to Determine
Can we find the exact total of all the turn angles?
Key Insight
Here's the crucial geometric principle: When you walk around the outside of any closed polygon and return to your starting point, the sum of all your turns (exterior angles) always equals \(360°\). This is true regardless of the polygon's shape or number of sides—it's a fundamental property of closed paths.
Think of it this way: If you face north and walk around any building, making turns at each corner, you'll end up facing north again. That means you've turned a total of \(360°\).
Analyzing Statement 1
Statement 1 tells us: One of the turns is \(80°\).
This gives us the measure of just one turn, but we don't know:
- How many sides the park has
- How many turns were made in total
- What the other turn angles are
Without knowing the shape or number of sides, we cannot determine the sum of all turns. For instance:
- In a triangle: 3 turns totaling \(360°\)
- In a hexagon: 6 turns totaling \(360°\)
- In any polygon: n turns totaling \(360°\)
Knowing one angle (\(80°\)) out of an unknown total doesn't help us find the sum.
Statement 1 is NOT sufficient.
✓ This eliminates choices A and D.
Analyzing Statement 2
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2 tells us: The park has 4 sides (quadrilateral), all sides are straight, and each interior angle is less than \(180°\) (making it convex).
This is exactly what we need! Since we now know:
- The park is a closed polygon (specifically a quadrilateral)
- It's convex (no inward-pointing angles)
We can apply our key geometric principle: The sum of exterior angles for any closed polygon equals \(360°\).
Therefore, the sum of all turns = \(360°\). [STOP - Sufficient!]
It doesn't matter if the quadrilateral is a square, rectangle, trapezoid, or any other 4-sided shape—the exterior angles always sum to \(360°\).
Statement 2 is sufficient.
✓ This eliminates choices C and E.
The Answer: B
Statement 2 alone gives us enough information to determine that the sum equals \(360°\), while Statement 1 alone does not.
Answer Choice B: "Statement 2 alone is sufficient, but Statement 1 alone is not sufficient."