Pythagorean Theorem for a 15-year-old: step-by-step, Desmos activities, AoPS practice, Wobbledogs & LEGO Spike Prime projects

Clear, age-appropriate explanation of the Pythagorean theorem with visual proofs, Desmos activities you can paste, practice problems and solutions, links to AoPS-style practice, and fun hands-on projects using Wobbledogs and LEGO Spike Prime.

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

Statement: In a right triangle (one angle = 90°), if the two legs are length a and b, and the hypotenuse (the side opposite the right angle) is length c, then

a² + b² = c²

What the parts mean

  • Legs: the two shorter sides that meet at the right angle — call them a and b.
  • Hypotenuse: the longest side, opposite the right angle — call it c.

Simple numeric example

If a = 3 and b = 4, then c = sqrt(3² + 4²) = sqrt(9 + 16) = sqrt(25) = 5. So a 3-4-5 triangle is a right triangle.

Two short intuitive proofs

1) Visual square proof (area idea)

Make a square of side (a+b). Inside it you can place 4 copies of the right triangle so that a central square of side c remains. The total area can be counted two ways:

  • Area of big square: (a + b)² = a² + 2ab + b².
  • Area as 4 triangles + middle square: 4*(1/2 a b) + c² = 2ab + c².

Equate them: a² + 2ab + b² = 2ab + c² → a² + b² = c².

2) Coordinate (algebraic) proof

Put the right triangle with the right angle at the origin, one leg along the x-axis (point B at (a,0)) and the other along the y-axis (point C at (0,b)). The distance from origin A=(0,0) to the opposite corner (a,b) is c. Distance formula: c = sqrt((a − 0)² + (b − 0)²) = sqrt(a² + b²). Squaring gives the theorem.

Try it on Desmos (interactive)

Open desmos.com/calculator and paste these expressions. Then move the sliders a and b to see the triangle and the computed c change.

// paste into Desmos as separate lines
a = 3
b = 4
A = (0,0)
B = (a,0)
C = (0,b)
segment(A,B)
segment(A,C)
segment(B,C)
c = sqrt(a^2 + b^2)
// draw the square on each leg (visualize areas)
polygon(A, B, (a,a), (0,a))   // square on side of length a
polygon(A, C, (b,b), (b,0))   // square on side of length b
// label point for hypotenuse length
text("c = " + c, (a/2, b/2))

This builds a right triangle with legs on the axes. Changing a and b updates c live. Try integer values to see Pythagorean triples (3,4,5), (5,12,13), etc.

Connections to AoPS Pre-Algebra / Intro to Algebra

  • AoPS Pre-Algebra and Intro to Algebra focus on the same ideas: working with squares, algebraic manipulation, and problem-solving. Search AoPS for problems titled "Pythagorean theorem", "distance formula", or "Pythagorean triples" for progressively harder problems.
  • Good practice: prove some triples are primitive (cannot be scaled down), or use algebra to find the missing side when one leg and the hypotenuse are given (solve for a: a = sqrt(c² − b²)).
  • AoPS community and Alcumus have challenge problems where you combine theorem use with factoring, number theory, or geometry — perfect for building contest-style reasoning.

Practice problems (with quick solutions)

  1. Legs 6 and 8 → hypotenuse c = sqrt(36 + 64) = sqrt(100) = 10.
  2. Hypotenuse 13, one leg 5 → missing leg = sqrt(13² − 5²) = sqrt(169 − 25) = sqrt(144) = 12.
  3. Right triangle coordinates (−2, 1) and (4, 5) are endpoints of a side; distance = sqrt((4 + 2)² + (5 − 1)²) = sqrt(6² + 4²) = sqrt(36 + 16) = sqrt(52) = 2*sqrt(13).

Deeper ideas (if you want to go further):

  • Converse: if a² + b² = c² for side lengths, the triangle is right-angled.
  • Generate Pythagorean triples: for integers m > n > 0, the triple (m² − n², 2mn, m² + n²) is a Pythagorean triple.
  • Distance formula in coordinate geometry is just the Pythagorean theorem applied to differences in coordinates.

Fun, hands-on projects using your interests

Wobbledogs (www.wobbledogs.com)

  • Challenge idea: while playing, look for triangular shapes created by limbs or accessories. Take screenshots and measure (in pixels) the horizontal and vertical distances between two points to test the Pythagorean theorem. Many image editors show pixel coordinates — the differences give you a and b; compute c and compare to direct pixel distance.
  • Creative math challenge: design a wobble-dog or build a pattern that relies on a 3:4:5 proportion. Use it as a design constraint and check visually whether the proportions produce a right angle.

LEGO Education Spike Prime

  • Build a right-triangle frame using Technic beams. Use beam lengths in a 3:4:5 ratio (e.g., 6 studs, 8 studs, 10 studs scaled up as needed) so the triangle is exactly right. Verify with a square corner or protractor.
  • Sensor-based test: program the Spike Prime robot to drive along leg a, then turn 90° and drive along leg b (measure using motor rotations). Compute expected hypotenuse length with the formula c = sqrt(a² + b²) in your Python or Scratch program, then have the robot attempt to drive that distance as a check. Compare measured wheel rotations to calculation — great practice in applying math and checking experimental error.
  • Math block example (MicroPython pseudo): 
    # assume a_cm and b_cm measured from rotations
c = (a_cm**2 + b_cm**2)**0.5
print('Expected hypotenuse (cm):', c)

Tips for studying

  • Memorize the formula a² + b² = c² and what each letter means, then practice recognizing right triangles in coordinate and word problems.
  • Use Desmos to build intuition: move sliders, see squares and areas change, and watch how c depends on a and b.
  • If you're preparing for contests (AoPS style), practice proofs (area and similarity) and problems that combine algebra and geometry.

If you want, I can:

  • Give more AoPS-style problems (with step-by-step solutions),
  • Create a ready-to-paste Desmos file (I'll give the exact lines again or provide a share link), or
  • Design a LEGO Spike Prime step-by-step build & program to measure the hypotenuse.

Which would you like next?

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