A right triangle has legs 9 and 12. Find the hypotenuse
视频信息
答案文本
视频字幕
We are given a right triangle with legs measuring 9 and 12 units. Our goal is to find the length of the hypotenuse. In a right triangle, the hypotenuse is the longest side, opposite the right angle. The two shorter sides are called legs.
To find the hypotenuse, we'll use the Pythagorean theorem. This fundamental theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. In mathematical terms, a squared plus b squared equals c squared, where a and b are the lengths of the legs, and c is the length of the hypotenuse.
Now we'll substitute our known values into the Pythagorean theorem. Our legs measure 9 and 12 units, so we replace a with 9 and b with 12 in the equation. This gives us 9 squared plus 12 squared equals c squared, where c represents our unknown hypotenuse.
Now let's calculate step by step. First, we compute 9 squared, which equals 81. Next, we compute 12 squared, which equals 144. Adding these together, 81 plus 144 equals 225. So c squared equals 225. To find c, we take the square root of 225, which equals 15.
We've determined that the hypotenuse is 15 units. To verify our solution, let's substitute back into the original equation. We check if 9 squared plus 12 squared equals 15 squared. Computing each term, we get 81 plus 144 equals 225, and 15 squared also equals 225. Since both sides are equal, our solution is confirmed. The sides 9, 12, and 15 form a Pythagorean triple.
The (9, 12, 15) triple is actually a multiple of the famous (3, 4, 5) Pythagorean triple. Each side of the (3, 4, 5) triangle has been multiplied by 3 to get our (9, 12, 15) triangle. Recognizing common Pythagorean triples like 3-4-5, 5-12-13, and 8-15-17 can significantly speed up problem solving in geometry and trigonometry.