Pythagorean Theorem Worksheet for 14-Year-Olds — Distance Word Problems

Instructions

Welcome to the Pythagorean Theorem Challenge! The Pythagorean theorem helps us find the missing side of a right-angled triangle. Remember the formula: a² + b² = c², where 'a' and 'b' are the two shorter sides (legs) that form the right angle, and 'c' is the longest side, called the hypotenuse.

For word problems like these, it's always a good idea to draw a simple diagram to visualize the situation. This will help you see the right-angled triangle you need to solve. Read each question carefully and show your work. Good luck!

Problems

1. W swims 60 miles north, then 30 miles east, then another 30 miles north, and finally 150 miles west. How far is W from the starting point? (Calculate the direct straight-line distance).
   
   Your work and answer here:
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?
   
   Your work and answer here:

Answer Key

1. Answer: W is 150 miles from the starting point.

   Explanation:

   1. First, combine the movements in the same or opposite directions.
      • Northward movement: 60 miles + 30 miles = 90 miles north.
      • East/West movement: 30 miles east and 150 miles west. This results in a net movement of 150 - 30 = 120 miles west.
   2. Now, you have a right-angled triangle. One leg is the total northward distance (a = 90 miles), and the other leg is the total westward distance (b = 120 miles). The distance from the start is the hypotenuse (c).
   3. Use the Pythagorean theorem: a² + b² = c²
      90² + 120² = c²
      8100 + 14400 = c²
      22500 = c²
      c = √22500
      c = 150 miles

2. Answer: B and G are approximately 104.4 meters apart.

   Explanation:

   This problem is best solved by mapping the points on a coordinate grid. Let's place point B at the origin (0, 0).

   1. Determine the East-West positions (the x-coordinate):
      • B is at x = 0.
      • A is 50m east of B → A is at x = 50.
      • C is 30m east of A → C is at x = 50 + 30 = 80.
      • D is 60m east of C → D is at x = 80 + 60 = 140.
      • E is 40m west of D → E is at x = 140 - 40 = 100.
   2. Determine the North-South positions (the y-coordinate):
      • We can assume B, A, C, D, and E are on the x-axis (y=0).
      • F is 50m north of E. Since E is at (100, 0), F is at (100, 50).
      • G is 80m south of F. Since F is at (100, 50), G is at (100, 50 - 80) which is (100, -30).
   3. Find the distance between B (0, 0) and G (100, -30).
      This forms a right-angled triangle.
      • The horizontal leg (a) is the change in x: 100 - 0 = 100m.
      • The vertical leg (b) is the change in y: 0 - (-30) = 30m.
   4. Use the Pythagorean theorem: a² + b² = c²
      100² + 30² = c²
      10000 + 900 = c²
      10900 = c²
      c = √10900
      c ≈ 104.403...
      Rounding to the nearest tenth gives 104.4m.