What is the Pythagorean Theorem formula (a² + b² = c²) for Grade 8?

📚 What is the Pythagorean Theorem?

The Pythagorean Theorem is a fundamental concept in geometry that describes the relationship between the sides of a right triangle. A right triangle is a triangle that has one angle that measures 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse, and the other two sides are called legs.

The theorem states: In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs). This relationship is expressed by the formula: $a^2 + b^2 = c^2$, where 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse.

📜 A Bit of History

The theorem is named after the ancient Greek mathematician Pythagoras. Although the theorem bears his name, evidence suggests that the relationship was known to earlier civilizations, such as the Babylonians and Egyptians. Pythagoras (c. 570 – c. 495 BC) is credited with providing the first proof of the theorem.

✨ Key Principles

• 📐 Right Triangle: The theorem only applies to right triangles, which have one angle of 90 degrees.
• 🦵 Legs: The two sides adjacent to the right angle are the legs (a and b).
• 🌟 Hypotenuse: The side opposite the right angle is the hypotenuse (c), which is also the longest side of the triangle.
• ➕ Sum of Squares: The sum of the squares of the legs ($a^2 + b^2$) is equal to the square of the hypotenuse ($c^2$).

➗ Applying the Formula: $a^2 + b^2 = c^2$

Let's look at how to use the formula in practice:

1. Identify the Right Triangle: Make sure the triangle has a right angle.
2. Label the Sides: Label the legs as 'a' and 'b', and the hypotenuse as 'c'.
3. Plug in the Values: Substitute the known values of 'a' and 'b' into the formula.
4. Solve for the Unknown: If you know 'a' and 'b', solve for 'c'. If you know 'a' and 'c', solve for 'b' (or vice versa).

➕ Real-World Examples

• 🪜 Ladder Against a Wall: Imagine a ladder leaning against a wall, forming a right triangle. The length of the ladder is the hypotenuse, the distance from the wall to the base of the ladder is one leg, and the height the ladder reaches on the wall is the other leg. If the ladder is 13 feet long and the base is 5 feet from the wall, you can find the height it reaches on the wall: $5^2 + b^2 = 13^2$. So, $25 + b^2 = 169$, $b^2 = 144$, and $b = 12$ feet.
• ⚾ Diagonal of a Rectangle: The diagonal of a rectangle divides it into two right triangles. If a rectangle is 4 inches wide and 3 inches long, the diagonal can be found using the Pythagorean Theorem: $3^2 + 4^2 = c^2$. So, $9 + 16 = c^2$, $25 = c^2$, and $c = 5$ inches.
• 🗺️ Navigation: A ship sails 3 miles east and then 4 miles north. How far is the ship from its starting point? This forms a right triangle with legs of 3 miles and 4 miles. Using the theorem: $3^2 + 4^2 = c^2$. So, $9 + 16 = c^2$, $25 = c^2$, and $c = 5$ miles.

📝 Practice Quiz

Here are some problems to test your understanding:

1. A right triangle has legs of length 6 cm and 8 cm. What is the length of the hypotenuse?
2. The hypotenuse of a right triangle is 10 inches long. One leg is 6 inches long. What is the length of the other leg?
3. A rectangular garden is 12 feet long and 5 feet wide. What is the length of the diagonal of the garden?

Answers:

1. 10 cm
2. 8 inches
3. 13 feet