Finding Shortest Distance with Pythagorean Theorem

What This Problem Teaches

• Recognizing when perpendicular movements form a right triangle
• Applying the Pythagorean theorem to find direct distances
• Simplifying square roots by factoring out perfect squares
• Understanding the difference between path distance and straight-line distance
• Converting between exact radical form and decimal approximations

Picture This

Mark's path creates a right triangle. The shortest distance home is the hypotenuse.

Solution: Method 1 — The Direct Pythagorean Approach

Step 1 — Identify the right triangle

Mark's path forms a perfect right triangle. He drives 35 miles North, then 42 miles West. Since North and West are perpendicular directions, these two segments are the legs of a right triangle, and the shortest distance home is the hypotenuse.

Step 2 — Set up the Pythagorean theorem

For any right triangle, the Pythagorean theorem states that c² = a² + b², where c is the hypotenuse and a and b are the legs.

Step 3 — Calculate the squares

Let's compute each term:

• 35² = 35 × 35 = 1,225
• 42² = 42 × 42 = 1,764

Step 4 — Add and take the square root

Step 5 — Simplify the square root

To simplify √2,989, we need to factor it. Let's check if it has any perfect square factors:

• 2,989 ÷ 49 = 61 (since 49 = 7²)
• So 2,989 = 49 × 61 = 7² × 61

For a decimal approximation: √61 ≈ 7.81, so c ≈ 7 × 7.81 = 54.67 miles.

Solution: Method 2 — Recognition of the 7-Factor Pattern

Step 1 — Notice the common factor

Both legs are multiples of 7:

• 35 = 7 × 5
• 42 = 7 × 6

Step 2 — Factor out the 7

We can rewrite the Pythagorean theorem as:

Step 3 — Take the square root directly

This method is faster when you recognize that both legs share a common factor, allowing you to factor before calculating large squares.

Verification

Let's verify our answer by substituting back into the Pythagorean theorem:

We can also check reasonableness: Mark's total path distance is 35 + 42 = 77 miles. The straight-line distance should be shorter, and indeed 54.7 < 77. Additionally, the hypotenuse should be longer than either leg, and 54.7 > 42 > 35 ✓.

Common Pitfalls

The Math Beneath the Surface

This problem demonstrates the fundamental principle that the shortest distance between two points is a straight line. When you have perpendicular movements (North-South and East-West), they form the legs of a right triangle, and the Pythagorean theorem gives you that straight-line distance.

The general formula for this type of problem is:

This same mathematical structure appears in:

• GPS navigation systems calculating "as the crow flies" distances
• Vector addition in physics (combining perpendicular forces)
• Distance formula in coordinate geometry: d = √[(x₂-x₁)² + (y₂-y₁)²]

Beyond the Textbook

This calculation appears constantly in real-world navigation:

• Aviation: Pilots calculate direct flight paths when air traffic control allows deviation from standard routes
• Emergency services: Fire departments and ambulances use straight-line distances to estimate response times
• Delivery logistics: Companies like Amazon use these calculations to optimize drone delivery routes

How to Spot This Problem Type

Look for these key phrases and setups:

• "Due North," "due South," "due East," or "due West" — these signal perpendicular movements
• "Shortest distance" or "direct distance" — this asks for the hypotenuse, not the path length
• Any movement in two perpendicular directions followed by a question about returning to the starting point
• "As the crow flies" — another way of asking for straight-line distance

If you see a problem involving two perpendicular movements and a question about distance, you're likely looking at a Pythagorean theorem problem.

Try These Variations