Pythagorean theorem for a 14-year-old — step-by-step guide, proofs, examples and real project use (PowerUp 2.0 Electric Paper Aeroplane, Beast Academy Ch12, AoPS Prealgebra)

What the Pythagorean theorem says

In any right-angled triangle (one angle is 90 degrees), the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. If the legs are a and b, and the hypotenuse is c, then:

a² + b² = c²

Why it works — two short proofs

1. Square-area proof (visual):

   Put four identical right triangles inside a square so the hypotenuses form a smaller tilted square inside. The big square area can be written two ways: as (a+b)², and as the area of the four triangles plus the area of the inner square (which has area c²). Setting them equal gives a² + b² = c².

2. Similar-triangles proof (algebraic):

   Drop the altitude from the right angle to the hypotenuse; that splits the triangle into two smaller triangles similar to the original. Using ratios from similarity and multiplying gives the same formula a² + b² = c².

How to use it — solve for the missing side

• Find the hypotenuse c when you know legs a and b: c = sqrt(a² + b²).
• Find a missing leg when you know c and the other leg b: a = sqrt(c² - b²). (Make sure c is the longest side; otherwise the triangle is impossible.)

Worked examples

1. Classic triple:

   a=6, b=8. Then c = sqrt(6² + 8²) = sqrt(36 + 64) = sqrt(100) = 10.

2. Find leg:

   c=13, b=5. Then a = sqrt(13² - 5²) = sqrt(169 - 25) = sqrt(144) = 12.

3. Paper size diagonal (real measurement):

   Suppose your paper is A4: 21.0 cm by 29.7 cm. The diagonal is c = sqrt(21.0² + 29.7²) = sqrt(441 + 882.09) = sqrt(1323.09) ≈ 36.38 cm. That tells you how long a straight fold from corner to corner will be.

Application: PowerUp 2.0 Electric Paper Aeroplane Conversion Kit

When you convert a paper plane with the kit, you sometimes need straight-line distances (for brace lengths, motor mount to tail, or center-of-mass checks). The Pythagorean theorem gives those distances when parts meet at right angles.

Example: you attach a motor mount that sits 18.0 cm along the fuselage from the tail and the mount is 2.5 cm above the fuselage centerline. What is the straight-line length of a support wire from the tail tip to the motor mount? Treat that as a right triangle with horizontal leg 18.0 cm and vertical leg 2.5 cm:

length = sqrt(18.0² + 2.5²) = sqrt(324 + 6.25) = sqrt(330.25) ≈ 18.17 cm.

This tells you the support wire should be cut slightly longer than 18.17 cm so it fits without stress. Always measure and add a little slack for fastening.

Distance formula (on the coordinate plane)

The Pythagorean theorem gives the distance between two points (x1,y1) and (x2,y2):

d = sqrt((x2 - x1)² + (y2 - y1)²)

Example: distance between (2,3) and (7,9) is sqrt(5² + 6²) = sqrt(61) ≈ 7.81.

Pythagorean triples

Some integers (a,b,c) satisfy a² + b² = c² exactly, like (3,4,5), (5,12,13), (8,15,17). These are useful because they give whole-number sides for right triangles (no square roots needed).

Practice problems (with quick answers)

1. Find c for a=9, b=12. (Answer: c = 15.)
2. Find the missing leg if c=25 and b=7. (Answer: a = sqrt(25² - 7²) = sqrt(625 - 49) = sqrt(576) = 24.)
3. Distance between (−1,4) and (3,−2). (Answer: sqrt(4² + 6²) = sqrt(52) ≈ 7.21.)

Common mistakes and tips

• Don’t use the formula unless the triangle is right-angled. If it isn’t, use the law of cosines.
• Always square the numbers first, then add or subtract, then take the square root last.
• Keep units consistent (cm with cm, m with m).
• If you get a negative number under the square root when solving for a leg, the triangle is impossible — check which side is the hypotenuse.

Curriculum links and further reading

• ACARA Grade 8–9: Applies to understanding and using the Pythagorean theorem to solve problems and to derive the distance formula.
• Beast Academy Ch12: A friendly, puzzle-based introduction that builds intuition with diagrams and practice.
• AoPS Prealgebra (Pythagorean sections): Deeper practice problems and theory for students who want a challenge.

Final note

The Pythagorean theorem is one of the most useful tools in geometry. It helps with measurements, building models (like your PowerUp plane), coordinate geometry and many contest problems. Practice with a few real and abstract problems and try to visualize the right triangle inside each situation.