Pythagorean Theorem Worksheet for 15-Year-Olds — Distance Word Problems (Geometry)

Instructions

Read each word problem carefully. Use the Pythagorean theorem (a² + b² = c²) to solve for the unknown distance. It may be helpful to draw a diagram for each problem to visualize the right-angled triangle. Show your work in the space provided.

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

   Show your work:

   
   
   
   
   

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

   Show your work:

   
   
   
   
   
   

Answer Key

1. Step 1: Calculate the total north-south distance.
   W travels 60 miles north + 30 miles north = 90 miles north. This is one leg of our triangle (a).

   Step 2: Calculate the total east-west distance.
   W travels 30 miles east and 150 miles west. The net movement is 150 - 30 = 120 miles west. This is the second leg of our triangle (b).

   Step 3: Use the Pythagorean theorem.
   a² + b² = c²
   90² + 120² = c²
   8100 + 14400 = c²
   22500 = c²
   c = √22500
   c = 150 miles

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

2. Step 1: Map the points on an East-West line.
   Let's set point B as our origin (0).
   - A is 50m east of B, so A is at 50m.
   - C is 30m east of A, so C is at 50 + 30 = 80m.
   - D is 60m east of C, so D is at 80 + 60 = 140m.
   - E is 40m west of D, so E is at 140 - 40 = 100m. So, point E is 100m east of point B. This is one leg of our triangle (a).

   Step 2: Map the points on a North-South line relative to the East-West line.
   - F is 50m north of E.
   - G is 80m south of F. This means G is 80 - 50 = 30m south of E. So, point G is 30m south of the horizontal line that B and E are on. This is the second leg of our triangle (b).

   Step 3: Use the Pythagorean theorem to find the distance between B (0m East, 0m North/South) and G (100m East, 30m South).
   a² + b² = c²
   100² + 30² = c²
   10000 + 900 = c²
   10900 = c²
   c = √10900
   c ≈ 104.403 meters

   Final Answer: To the nearest tenth of a meter, B and G are 104.4 meters apart.