Pythagorean Theorem Worksheet for 15-Year-Olds — Desmos Graphing + AoPS Pre‑Algebra & Intro to Algebra with LEGO SPIKE Prime

Instructions

Complete the following sections to practice and apply your knowledge of the Pythagorean theorem. You may use a calculator for your computations. For Part 2, you will need to use the Desmos online graphing calculator.

Part 1: The Basics - A Quick Refresher

For a right triangle with legs a and b and hypotenuse c, the Pythagorean theorem states: a² + b² = c². Use this to solve the following problems. Round answers to one decimal place where necessary.

1. A right triangle has legs of length 8 cm and 15 cm. What is the length of the hypotenuse?
2. A right triangle has a hypotenuse of 25 inches and one leg of 7 inches. What is the length of the other leg?
3. Do side lengths of 9 m, 12 m, and 15 m form a right triangle? Explain why or why not, using the theorem.

Part 2: Graphing the Distance with Desmos

The Pythagorean theorem is the foundation of the distance formula in algebra. Let's use it to find the straight-line distance between two points on a coordinate plane.

1. Go to www.desmos.com/calculator.
2. Plot the following two points: Point A = (-2, -1) and Point B = (6, 5). You can do this by typing `(-2, -1)` and `(6, 5)` into the expression list.
3. Imagine a right triangle where the distance between A and B is the hypotenuse. The horizontal distance between the points forms one leg, and the vertical distance forms the other.
   • What is the length of the horizontal leg (the change in x-values)?
   • What is the length of the vertical leg (the change in y-values)?
4. Use the Pythagorean theorem with the leg lengths you just found to calculate the exact distance between Point A and Point B.

Part 3: The Wobbledogs Conundrum

You've bred a prize-winning Wobbledog with a very, very long body. When it lays down perfectly straight to sleep, its body measures exactly 50 units from nose to tail. You are designing a custom rectangular pen for it.

• Scenario A: The pen is 48 units long and 14 units wide. Will your Wobbledog be able to fit if it sleeps perfectly along the diagonal of the pen? Show your work.
• Scenario B: You want to build a different pen where the Wobbledog can just fit along the diagonal. If you make the pen 30 units wide, what is the minimum length the pen would need to be?

Part 4: LEGO® SPIKE™ Prime Robotics Path

Your SPIKE Prime robot is on a large floor grid. It starts at position (0, 0). Its goal is to retrieve a LEGO brick located at position (12, 16), where each unit on the grid represents one centimeter.

1. To get the brick, the robot can only make 90-degree turns (it can only move horizontally and vertically). What is the total distance the robot must travel along the grid lines to reach the brick?
2. What is the straight-line, "as-the-crow-flies" distance from the start to the brick?
3. Challenge: The medium motor wheels on the SPIKE Prime set have a circumference of approximately 17.6 cm. How many full wheel rotations would the robot need to complete to travel the straight-line distance you calculated in question 2? (Round your final answer to the nearest whole rotation).

Part 5: AoPS Pre-Algebra/Algebra Challenge

This problem requires thinking in three dimensions. Imagine a rectangular box (like a shoebox). You want to find the longest straight object that can fit inside. This is called the "space diagonal," which connects opposite corners of the box.

Your box has a length of 12 cm, a width of 5 cm, and a height of 4 cm. What is the length of the space diagonal?

Hint: Use the Pythagorean theorem twice! First, find the diagonal of the bottom face of the box. Then, use that diagonal as a leg of a new right triangle, with the box's height as the other leg.

Answer Key

Part 1: The Basics - A Quick Refresher

1. 17 cm. (a² + b² = c² → 8² + 15² = c² → 64 + 225 = c² → 289 = c² → c = 17)
2. 24 inches. (a² + b² = c² → 7² + b² = 25² → 49 + b² = 625 → b² = 576 → b = 24)
3. Yes, it is a right triangle. Because 9² + 12² = 15². (81 + 144 = 225, and 15² is 225. Since a² + b² = c², it satisfies the theorem.)

Part 2: Graphing the Distance with Desmos

1. (Completed on Desmos)
2. (Completed on Desmos)
3. • Horizontal leg (change in x): 6 - (-2) = 8 units.
   • Vertical leg (change in y): 5 - (-1) = 6 units.
4. The distance is 10 units. (a² + b² = c² → 8² + 6² = c² → 64 + 36 = c² → 100 = c² → c = 10)

Part 3: The Wobbledogs Conundrum

• Scenario A: Yes, it will fit. The diagonal of the pen is 50 units. (48² + 14² = c² → 2304 + 196 = c² → 2500 = c² → c = 50). Since the diagonal is exactly 50 units, the Wobbledog will fit perfectly.
• Scenario B: The pen needs to be 40 units long. (The Wobbledog's length of 50 is the hypotenuse). (a² + b² = c² → a² + 30² = 50² → a² + 900 = 2500 → a² = 1600 → a = 40).

Part 4: LEGO® SPIKE™ Prime Robotics Path

1. The total distance is 28 cm. (The robot must travel 12 cm horizontally and 16 cm vertically. 12 + 16 = 28 cm).
2. The straight-line distance is 20 cm. (12² + 16² = c² → 144 + 256 = c² → 400 = c² → c = 20).
3. 1 rotation. (Total Distance / Circumference = Number of Rotations → 20 cm / 17.6 cm ≈ 1.136 rotations. Since the question asks for full rotations, the answer is 1).

Part 5: AoPS Pre-Algebra/Algebra Challenge

The length of the space diagonal is approximately 13.6 cm.

Step 1: Find the diagonal of the bottom face (length=12, width=5).

12² + 5² = d² → 144 + 25 = d² → 169 = d² → d = 13 cm.

Step 2: Use that diagonal (d=13) and the height (h=4) as the legs of a new triangle to find the space diagonal (s).

d² + h² = s² → 13² + 4² = s² → 169 + 16 = s² → 185 = s² → s ≈ 13.6 cm.