Pythagorean theorem proof using similarity | Geometry | Khan Academy

TL;DR

The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.

Transcript

This triangle that we have right over here is a right triangle. And it's a right triangle because it has a 90 degree angle, or has a right angle in it. Now, we call the longest side of a right triangle, we call that side, and you could either view it as the longest side of the right triangle or the side opposite the 90 degree angle, it is called a ... Read More

Key Insights

• 🔺 A right triangle has a 90-degree angle and a hypotenuse opposite the right angle.
• 🗯️ The Pythagorean Theorem states that a^2 + b^2 = c^2 in a right triangle.
• 🔺 Similar triangles within the right triangle can be used to prove the Pythagorean Theorem.
• 🗯️ The theorem is widely used in geometry and trigonometry to find unknown side lengths in right triangles.
• 📛 It is one of the most famous theorems in mathematics and is named after Pythagoras.
• 💁 The Pythagorean Theorem forms the basis of many geometric and trigonometric concepts.
• 🗯️ The theorem is applicable to any right triangle, regardless of its size or proportions.

Questions & Answers

Q: What is the Pythagorean Theorem?

The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two shorter sides (a and b) is equal to the square of the hypotenuse (c).

Q: How is similarity used to prove the Pythagorean Theorem?

By establishing similarity between triangles within the right triangle, we can set up ratios of corresponding sides and ultimately derive the relationship a^2 + b^2 = c^2.

Q: Why is the Pythagorean Theorem important?

The Pythagorean Theorem is fundamental in geometry and trigonometry, allowing us to find the length of an unknown side in a right triangle when the lengths of the other two sides are known.

Q: Who is the Pythagorean Theorem named after?

The theorem is named after the mathematician Pythagoras, although it is not clear if he was the first to discover it.

Summary & Key Takeaways

• A right triangle has a 90-degree angle and the longest side opposite the right angle is called the hypotenuse.

• The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

• By establishing similarity between triangles within the right triangle, it can be shown that a^2 + b^2 = c^2.