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In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
- Find the length of the hypotenuse. Our goal is to solve for the length of the hypotenuse. We are given the lengths of the two legs. We know two sides out of the three!
- Find the length of the leg. Just by looking at the figure above, we know that we have enough information to solve for the missing side. The reason is the measure of the two sides are given and the other leg is left as unknown.
- Do the sides [latex]17[/latex], [latex]15[/latex] and [latex]8[/latex] form a right triangle? If so, which sides are the legs and the hypotenuse?
- A rectangle has a length of [latex]8[/latex] meters and a width of [latex]6[/latex] meters. What is the length of the diagonal of the rectangle?
Learn how to use and prove the Pythagorean theorem, a special relationship between the sides of a right triangle. Explore examples, word problems, distance formula, and proofs with interactive exercises and quizzes.
- Why Is This Useful?
- How Do I Use It?
- And You Can Prove The Theorem Yourself !
- Another, Amazingly Simple, Proof
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If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!)
Write it down as an equation: Then we use algebrato find any missing value, as in these examples: Read Builder's Mathematicsto see practical uses for this. Also read about Squares and Square Roots to find out why √169 = 13 It works the other way around, too: when the three sides of a triangle make a2 + b2 = c2, then the triangle is right angled.
Get paper pen and scissors, then using the following animation as a guide: 1. Draw a right angled triangle on the paper, leaving plenty of space. 2. Draw a square along the hypotenuse (the longest side) 3. Draw the same sized square on the other side of the hypotenuse 4. Draw lines as shown on the animation, like this: 5. Cut out the shapes 6. Arra...
Here is one of the oldest proofs that the square on the long side has the same area as the other squares. Watch the animation, and pay attention when the triangles start sliding around. You may want to watch the animation a few times to understand what is happening. The purple triangle is the important one. We also have a proof by adding up the are...
Learn the formula, definition, examples and proof of Pythagoras' Theorem, which relates the lengths of the sides of a right angled triangle. Explore the history, applications and extensions of this famous mathematical result.
Learn the definition, formula, and applications of the Pythagorean theorem, which relates the lengths of the sides of a right triangle. Explore examples, exercises, and interactive diagrams on this topic.
May 10, 2024 · Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a 2 + b 2 = c 2.
Learn how to use the Pythagorean theorem to find the missing side length of a right triangle. Watch a video, see examples, and read comments from other learners.
- 11 min
- Sal Khan
