TEOREMA DE PITÁGORAS

Pythagorean theorem (video)

Main ideas and concepts

• Definition: A right triangle has one 90° angle. The two sides that meet at the right angle are the legs; the side opposite the right angle (the longest side) is the hypotenuse.

• Theorem (for right triangles):

  If a and b are the legs and c is the hypotenuse, then a^2 + b^2 = c^2.

• Geometric interpretation: The area of the square built on the hypotenuse equals the sum of the areas of the squares built on the two legs (visual/area-proof idea).

• Practical usage — two main problem types:
  • Given both legs, find the hypotenuse.
  • Given the hypotenuse and one leg, find the other leg.
• Tip: Always first identify the right angle so you know which sides are legs and which is the hypotenuse. Letters a, b, c are just labels — what matters is which side is opposite the right angle.

Methodologies / step-by-step procedures

1) Find the hypotenuse c when both legs a and b are known

Steps:

1. Write the theorem: a^2 + b^2 = c^2.
2. Substitute the known leg lengths for a and b.
3. Compute a^2 and b^2, add them to get c^2.
4. Take the square root: c = sqrt(a^2 + b^2).

Example (legs 3 cm and 4 cm):

• 3^2 + 4^2 = 9 + 16 = 25
• c = sqrt(25) = 5 cm

2) Find a missing leg (say a) when hypotenuse c and the other leg b are known

Steps:

1. Start from a^2 + b^2 = c^2 and solve for a^2: a^2 = c^2 − b^2.
2. Substitute the known values for c and b.
3. Compute c^2 and b^2, subtract to get a^2.
4. Take the square root: a = sqrt(c^2 − b^2).

Example 1:

• c = 5, b = 4
• a^2 = 5^2 − 4^2 = 25 − 16 = 9
• a = sqrt(9) = 3

Example 2:

• c = 15 cm, b = 12 cm
• a^2 = 15^2 − 12^2 = 225 − 144 = 81
• a = sqrt(81) = 9 cm

3) Word-problem application (ladder problem)

• Interpretation: ladder length = hypotenuse, wall height = one leg, distance from wall = the other leg.
• Identify which lengths are given and which is unknown, then use the appropriate method above.

Example:

• A 10 m ladder leans with its foot 8 m from the wall.
• c = 10, one leg = 8 → other leg = sqrt(10^2 − 8^2) = sqrt(100 − 64) = sqrt(36) = 6 m

Key takeaways / tips

• The theorem only applies to right triangles.
• Always identify the right angle first to label legs and the hypotenuse correctly.
• Process summary:
  • To find c: add squares of legs, then take the square root.
  • To find a leg: subtract square of the known leg from square of hypotenuse, then take the square root.
• Letters (a, b, c) are labels — adapt them to whichever sides are legs and hypotenuse in the problem.

Speakers / sources featured

• Channel presenter / narrator (teacher explaining the Pythagorean theorem)
• Historical source mentioned: Pythagoras