Pythagorean Theorem Worksheet for 15-Year-Olds — LEGO Education SPIKE Prime STEM Math

Instructions

Imagine you are building with your LEGO® Education SPIKE™ Prime kit. The length of the beams and the distance your robot travels are often measured in "studs" or centimeters. Use the Pythagorean theorem to solve the following building and programming challenges. Remember to show your work. Round decimal answers to the nearest hundredth if necessary.

Quick Refresher: The Pythagorean Theorem
For any right-angled triangle, the square of the hypotenuse (the side opposite the right angle, 'c') is equal to the sum of the squares of the other two sides ('a' and 'b').

a² + b² = c²

Exercises

1. You are building a support brace for a robotic arm. You use a vertical beam that is 12 studs long and a horizontal beam that is 16 studs long. They meet at a perfect 90° angle. What is the minimum length (in studs) of a diagonal beam needed to connect the two ends?



2. Your robot has a grabbing claw mounted on a tall vertical beam. A slanted support beam that is 25 studs long (the hypotenuse) connects the top of the vertical beam to a point on the chassis that is 7 studs away from the base of the vertical beam. How tall is the vertical beam?



3. You program your robot to move from its starting position. It travels 60 cm forward, makes a perfect 90° right turn, and then travels 11 cm. What is the straight-line distance from your robot's starting point to its final position?



4. To ensure your robot's chassis is perfectly square, you decide to check a corner. You have built a triangle in the corner using beams that are 8 studs, 15 studs, and 17 studs long. If the 8-stud and 15-stud beams are intended to form the right angle, does this combination of beams create a perfect 90° corner? Use the converse of the Pythagorean theorem to explain why or why not.



5. Challenge: You want to build a large rectangular frame for a project. The frame has a height of 20 studs and a width of 21 studs. To make it rigid, you need to add two diagonal cross-beams from corner to corner. What is the length of one of those diagonal beams?



6. Advanced Challenge: You've built a box-shaped structure with a base of 12 studs by 16 studs, and a height of 15 studs. What is the longest straight axle that could fit inside the box, reaching from one bottom corner to the opposite top corner?
   (Hint: Use the 3D Pythagorean theorem, d² = a² + b² + c², where a, b, and c are the length, width, and height).

Answer Key

1. Answer: 20 studs
   Work: a² + b² = c² → 12² + 16² = c² → 144 + 256 = c² → 400 = c² → c = √400 = 20.
2. Answer: 24 studs
   Work: a² + b² = c² → a² + 7² = 25² → a² + 49 = 625 → a² = 625 - 49 → a² = 576 → a = √576 = 24.
3. Answer: 61 cm
   Work: a² + b² = c² → 60² + 11² = c² → 3600 + 121 = c² → 3721 = c² → c = √3721 = 61.
4. Answer: Yes, it creates a perfect 90° corner.
   Explanation: According to the converse of the Pythagorean theorem, if a² + b² = c², then the triangle is a right-angled triangle. We check if 8² + 15² = 17².
   64 + 225 = 289.
   17² = 289. Since both sides of the equation are equal, the beams form a perfect right angle. This is a Pythagorean triple (8-15-17).
5. Answer: 29 studs
   Work: The diagonal is the hypotenuse. a² + b² = c² → 20² + 21² = c² → 400 + 441 = c² → 841 = c² → c = √841 = 29.
6. Answer: 25 studs
   Work: d² = a² + b² + c² → d² = 12² + 16² + 15² → d² = 144 + 256 + 225 → d² = 400 + 225 → d² = 625 → d = √625 = 25.