Pythagorean Theorem — a clear, step-by-step guide for a 15-year-old

Goal: Understand what the theorem says, why it's true, how to use it to solve problems, and how to test it yourself with hands-on projects using LEGO Spike Prime and Wobbledogs. I also point you to AoPS Pre-Algebra and Intro to Algebra ideas for extra practice.

1. Statement (the short version)

In any right triangle (one angle is 90 degrees), let the legs (the two sides that form the right angle) be a and b, and let the hypotenuse (the side opposite the right angle) be c. Then:

a² + b² = c²

2. What it means (in plain language)

• Square the lengths of the two shorter sides and add them; the result equals the square of the longest side.
• Example: If a = 3 and b = 4, then 3² + 4² = 9 + 16 = 25, so c = sqrt(25) = 5.

3. Simple visual proof (area rearrangement)

1. Make a big square of side (a + b).
2. Inside it place four copies of the same right triangle (legs a and b, hypotenuse c) with the triangles arranged so their hypotenuses form a small central square of side c.
3. Area of the big square = (a + b)² = a² + 2ab + b².
4. Area counted another way: 4 triangles (each area 1/2 ab) plus the central square (area c²) gives 2ab + c².
5. Set them equal: a² + 2ab + b² = 2ab + c². Cancel 2ab to get a² + b² = c².

This is a visual and very accessible proof you can draw or build with real triangles.

4. Converse

If for a triangle the side lengths satisfy a² + b² = c², then the triangle is a right triangle (with c as the hypotenuse).

5. How to use it (step-by-step solving)

1. Identify which side is the hypotenuse (longest side). Call the legs a and b and the hypotenuse c.
2. If you need c: compute c = sqrt(a² + b²).
3. If you need a (given b and c): compute a = sqrt(c² - b²).
4. Remember to check units and make sure the number under the square root is non-negative.

6. Examples with solutions

Example 1 (classic): a = 3, b = 4. c = sqrt(3² + 4²) = sqrt(9 + 16) = sqrt(25) = 5.

Example 2 (solve for a): b = 6, c = 10. Then a = sqrt(10² - 6²) = sqrt(100 - 36) = sqrt(64) = 8.

Example 3 (word problem): A ladder reaches a window 12 ft above the ground. The base of the ladder is 5 ft from the wall. How long is the ladder? c = sqrt(12² + 5²) = sqrt(144 + 25) = sqrt(169) = 13 ft.

7. Pythagorean triples

Integers (a, b, c) satisfying the theorem are called Pythagorean triples. Examples: (3,4,5), (5,12,13), (8,15,17). You can scale a triple by any positive integer (for example 6,8,10 is 2 times 3,4,5).

8. Where AoPS (Pre-Algebra and Intro to Algebra) helps

• AoPS Pre-Algebra has lots of practice on basic number properties, operations with squares and square roots, and integer patterns (useful for spotting triples).
• AoPS Intro to Algebra develops equation solving and manipulating square roots; those skills let you solve for missing sides and handle word problems confidently.
• Try problem sets that mix algebra with geometry: solve for unknown side lengths, prove something is a right triangle, or work with similar triangles that lead to Pythagorean-type equations.

9. Hands-on projects you can try

A. LEGO Education Spike Prime: build and measure

1. Build a rigid right-triangle frame using beams for the two legs and one beam for the hypotenuse. Make sure joints are fixed so lengths are accurate.
2. Measure the leg lengths in LEGO units (studs or centimeters) precisely. Compute a² + b² and take the square root to predict c.
3. If you use a distance sensor or wheel encoders you can program Spike Prime to compute a² + b² and compare to measured c². Example program steps:
   • Read numeric lengths for a, b, and c (either by manual input or sensor readings).
   • Compute left = a*a + b*b and right = c*c.
   • Print both values and show if left equals right (allowing small tolerance for measurement error).
4. Challenge: program the robot to move along one leg, then the other, and finally drive straight to the opposite corner; compare actual travel distance to predicted hypotenuse.

B. Wobbledogs: visualize triangles and measure

Wobbledogs makes creatures with joints connected by bones. You can:

• Find three joints that form a triangle (for example, shoulder, hip, and paw). Take a screenshot.
• Open the screenshot in an image editor and use the pixel-measure tool to measure distances between the joint centers (these are proportional to actual lengths).
• Pick two shorter sides as a and b and the longest as c; test whether a² + b² ≈ c². Because of perspective and screen pixels you will get approximate equality — but it's a fun way to explore geometry in a game.

10. Practice problems (try these) with answers

1. Find c if a = 7 and b = 24. Answer: c = sqrt(7² + 24²) = sqrt(49 + 576) = sqrt(625) = 25.
2. Given c = 13 and b = 5, find a. Answer: a = sqrt(13² - 5²) = sqrt(169 - 25) = sqrt(144) = 12.
3. Is a triangle with sides 6, 8, 10 a right triangle? Check: 6² + 8² = 36 + 64 = 100 = 10², so yes.
4. Word problem: A TV is 52 inches diagonally and its height is 30 inches. How wide is it? Let c = 52, a = 30, find b. b = sqrt(52² - 30²) = sqrt(2704 - 900) = sqrt(1804) ≈ 42.47 inches.

11. Tips and common mistakes

• Make sure you identify the hypotenuse correctly (it is always the longest side).
• Don’t mix up which sides are a, b, c — algebraically you can rename legs, but c must be the hypotenuse.
• Watch out for rounding errors when measuring with sensors or pixels; allow a small tolerance when you compare numbers.
• Remember the theorem only holds for right triangles. For non-right triangles use the law of cosines instead.

12. Next steps

• Work AoPS Pre-Algebra problems about square roots and integer properties to become comfortable with squares and square roots.
• Move to AoPS Intro to Algebra problems that mix algebra and geometry — solving equations and manipulating radicals.
• Try more hands-on experiments with Spike Prime: program it to test many random triangular frames and log the results.

If you want, I can:

• Give a specific Spike Prime program (pseudocode or Python) to compute and compare a² + b² with c².
• Make 5 AoPS-style practice problems of increasing difficulty with step-by-step solutions.
• Create a step-by-step photo plan for a Spike Prime build that forms a right triangle of known leg lengths.

Which of those would you like next?