12. A leg of a right triangle is 30 meters long, and the hypotenuse is 35 meters long. What is the length of the other leg?

To find the length of the other leg in a right triangle, you can use the Pythagorean theorem, which states that \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse and \( a \) and \( b \) are the legs. Here, you have one leg \( a = 30 \) meters and the hypotenuse \( c = 35 \) meters. So, rearranging the formula, we have: \( b^2 = c^2 - a^2 \) Substituting the values: \( b^2 = 35^2 - 30^2 \) Calculating the squares gives: \( b^2 = 1225 - 900 \) \( b^2 = 325 \) Now take the square root to find \( b \): \( b = \sqrt{325} \approx 18.03 \) meters. The length of the other leg is approximately 18.03 meters. --- Did you know that the Pythagorean theorem isn't just a mathematical principle but a fundamental concept that has been used since ancient times? It was named after the Greek mathematician Pythagoras, who lived around 570-495 BCE, and while he didn’t originate the theorem, he is credited with its popularization. Ancient cultures, including the Babylonians and Indians, also possessed similar knowledge long before Pythagoras! In real-world applications, the Pythagorean theorem is super useful! Architects and engineers use it all the time for structural designs, ensuring everything is up to code while maintaining balance and stability in buildings. It's also handy in navigation, allowing sailors and pilots to calculate distances accurately when plotting courses! So, the next time you see a building or a ship set sail, remember Pythagoras had a hand in that, centuries ago!