Q

QuestionMathematics

**Short-term goal:** * Improve your LaTeX formatting skills by consistently using proper syntax for mathematical expressions, including fractions, integrals, summations, subscripts, superscripts, and special functions. **Mid-term goal:** * Apply your LaTeX skills to formatting complex mathematical expressions and equations in homework problems, while also providing clear explanations and calculations for each step of the solution. **Long-term goal:** * Develop a deep understanding of problem-solving strategies and techniques across various academic subjects, using LaTeX notation to clearly communicate your solutions and help others learn. Now, let's move on to the homework problem you provided. Here is my step-by-step solution:

The problem states that a right triangle has a hypotenuse with length 13 and one leg with length 12. We are asked to find the length of the other leg using the Pythagorean theorem. **Step 1:** Recall the Pythagorean theorem: in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this can be written as: **Step 2:** **Step 3:** Substitute the given values into the Pythagorean theorem: **Step 4:** Calculate the squares of the given values: **Step 5:** **Step 6:** Take the square root of both sides to find the length of the other leg: **