๐ 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 with one angle that measures 90 degrees. The side opposite the right angle is called the hypotenuse (c), and the other two sides are called legs (a and b).
The theorem states: $a^2 + b^2 = c^2$, where 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse. In simpler terms, the sum of the squares of the legs is equal to the square of the hypotenuse.
๐ History of the Pythagorean Theorem
While named after the ancient Greek mathematician Pythagoras, evidence suggests that the relationship was known to earlier civilizations, such as the Babylonians and Egyptians. Pythagoras (c. 570 โ c. 495 BC) is credited with providing the first formal proof of the theorem.
๐ Key Principles
- ๐ Right Triangles Only: The Pythagorean Theorem only applies to right triangles.
- โ Legs and Hypotenuse: It relates the lengths of the two legs (a and b) to the length of the hypotenuse (c).
- ๐งฎ Formula: $a^2 + b^2 = c^2$ is the core equation.
๐ Real-World Examples
Example 1: Finding the Hypotenuse
Imagine a right triangle with legs of length 3 and 4. Find the length of the hypotenuse.
Solution:
Using the formula: $a^2 + b^2 = c^2$
$3^2 + 4^2 = c^2$
$9 + 16 = c^2$
$25 = c^2$
$c = \sqrt{25} = 5$
The hypotenuse is 5.
Example 2: Finding a Leg
Suppose a right triangle has a hypotenuse of length 13 and one leg of length 5. Find the length of the other leg.
Solution:
Using the formula: $a^2 + b^2 = c^2$
$5^2 + b^2 = 13^2$
$25 + b^2 = 169$
$b^2 = 169 - 25$
$b^2 = 144$
$b = \sqrt{144} = 12$
The other leg is 12.
Example 3: A Ladder Against a Wall
A 10-foot ladder is leaning against a wall. The base of the ladder is 6 feet away from the wall. How high up the wall does the ladder reach?
Solution:
Here, the ladder is the hypotenuse (c = 10), and the distance from the wall is one leg (a = 6). We need to find the height up the wall (b).
$a^2 + b^2 = c^2$
$6^2 + b^2 = 10^2$
$36 + b^2 = 100$
$b^2 = 100 - 36$
$b^2 = 64$
$b = \sqrt{64} = 8$
The ladder reaches 8 feet up the wall.
๐ Practice Quiz
Question 1
A right triangle has legs of length 8 and 15. What is the length of the hypotenuse?
Question 2
The hypotenuse of a right triangle is 17, and one leg is 8. Find the length of the other leg.
Question 3
A baseball diamond is a square with sides of 90 feet. What is the distance from home plate to second base (the diagonal of the square)?
Question 4
A ramp is 5 meters long and rises to a height of 1 meter. What is the horizontal distance covered by the ramp?
Question 5
A television screen is 36 inches wide and 27 inches tall. What is the length of its diagonal?
Question 6
A garden gate is 4 feet wide and 3 feet high. What is the length of a diagonal brace needed to support the gate?
Question 7
An airplane flies 9 miles east and then 12 miles north. What is the straight-line distance from its starting point?
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Conclusion
The Pythagorean Theorem is a powerful tool for solving problems involving right triangles. Understanding its principles and practicing with examples can help you master this important concept in geometry. Good luck!