Pythagorean Theorem Worksheet for 23-Year-Olds — Distance & Navigation Problems (2 Exercises)

Instructions

Read each problem carefully. It is highly recommended to draw a diagram for each scenario to visualize the movements and distances. Use the Pythagorean theorem (a² + b² = c²) to find the unknown distance. For the final answers, round to the specified decimal place where required.

Exercises

1. W swims 60 miles north, 30 miles east, 30 miles north, and finally 150 miles west. After all this travel, how far is W from the original starting point?

   
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2. A is 50m east of B and 30m west of C. D is 60m east of C, and 40m east of E. F is 50m north of E and 80m north of G. To the nearest tenth of a meter, how far apart are B and G?

   
   Show your work here:
   
   
   
   
   

Answer Key

1. Answer: 150 miles

   Explanation:

   • First, combine the north-south movements and the east-west movements.
   • Total Northward movement: 60 miles + 30 miles = 90 miles North.
   • Total East-West movement: 30 miles East - 150 miles West = -120 miles, which means 120 miles West.
   • This creates a right-angled triangle with the two legs being 90 miles and 120 miles. The distance from the start is the hypotenuse.
   • Using the Pythagorean theorem: a² + b² = c²
   • 90² + 120² = c²
   • 8100 + 14400 = c²
   • 22500 = c²
   • c = √22500
   • c = 150 miles

2. Answer: 104.4 m

   Explanation:

   • Establish a coordinate system, placing point B at the origin (0, 0).
   • Determine horizontal (East-West) positions:
   • B is at (0, 0).
   • A is 50m east of B, so A is at (50, 0).
   • C is 30m east of A, so C is at (50 + 30, 0) = (80, 0).
   • D is 60m east of C, so D is at (80 + 60, 0) = (140, 0).
   • E is 40m west of D, so E is at (140 - 40, 0) = (100, 0).
   • Determine vertical (North-South) positions relative to E:
   • F is 50m north of E, so F is at (100, 50).
   • F is 80m north of G. This means G is 80m south of F. So G's y-coordinate is 50 - 80 = -30. G's x-coordinate is the same as E and F's. So, G is at (100, -30).
   • Calculate the distance between B (0, 0) and G (100, -30). This distance is the hypotenuse of a right triangle.
   • The horizontal leg (a) is the difference in x-coordinates: 100 - 0 = 100m.
   • The vertical leg (b) is the difference in y-coordinates: 0 - (-30) = 30m.
   • Using the Pythagorean theorem: 100² + 30² = c²
   • 10000 + 900 = c²
   • 10900 = c²
   • c = √10900 ≈ 104.403...
   • Rounding to the nearest tenth, the distance is 104.4 m.