A right triangle has legs of length 9 cm and 12 cm. What is the length of the altitude to the hypotenuse?
["Right Triangle with Legs 9 cm and 12 cm: Find the Altitude to the Hypotenuse", "If you’re learning geometry, one of the most practical applications of right triangles is calculating the altitude to the hypotenuse. In this article, we’ll explore a compelling problem: a right triangle with legs measuring 9 cm and 12 cm and determine the exact length of the altitude drawn to its hypotenuse.", "---", "### Understanding the Problem", "A right triangle has two legs and a hypotenuse — the longest side opposite the right angle. Given leg lengths of 9 cm and 12 cm, we want to find the altitude (a perpendicular line) from the right angle to the hypotenuse.", "This altitude plays a key role in various areas of geometry and real-world applications, such as architecture, physics, and computer graphics. Understanding how to calculate it strengthens your grasp of triangle properties and area relationships.", "---", "### Step 1: Compute the Hypotenuse Length", "Using the Pythagorean theorem:\n[\nc = \sqrt{a^2 + b^2}\n]", "Where ( a = 9 ) cm, ( b = 12 ) cm.\n[\nc = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \ ext{ cm}\n]", "So, the hypotenuse has a length of 15 cm.", "---", "### Step 2: Use the Area to Find the Altitude", "An efficient way to find the altitude ( h ) to the hypotenuse is by comparing the triangle’s area calculated in two ways:", "1. Using the legs:\n[\n\ ext{Area} = \frac{1}{2} \ imes a \ imes b = \frac{1}{2} \ imes 9 \ imes 12 = 54 \ ext{ cm}^2\n]", "2. Using the hypotenuse and the altitude:\n[\n\ ext{Area} = \frac{1}{2} \ imes c \ imes h = \frac{1}{2} \ imes 15 \ imes h\n]", "Since both expressions represent the same area, we equate them:", "[\n\frac{1}{2} \ imes 15 \ imes h = 54\n]", "Multiply both sides by 2:", "[\n15h = 108\n]", "Solve for ( h ):", "[\nh = \frac{108}{15} = 7.2 \ ext{ cm}\n]", "---", "### Step 3: Verify with Trigonometric Relationship (Optional Insight)", "The ratio of each leg to the hypotenuse corresponds to sine values:\n[\n\sin(\ heta) = \frac{9}{15} = \frac{3}{5},\quad \sin(\phi) = \frac{12}{15} = \frac{4}{5}\n]", "These angles (36.87° and 53.13°) have nice ratios supporting the altitude calculation in right triangles. The derived altitude ( h = 7.2 ) cm aligns with standard triangle altitude formulas.", "---", "### Final Answer", "The length of the altitude to the hypotenuse in a right triangle with legs 9 cm and 12 cm is 7.2 cm.", "---", "### Why This Matters", "Calculating the altitude to the hypotenuse strengthens understanding of triangle area and proportional relationships. It’s crucial for solving more complex geometric problems, determining triangle characteristics, and applying formulas in technical fields.", "---", "Keywords: right triangle altitude, legs 9 cm and 12 cm, hypotenuse altitude, right triangle area calculation, geometry problem solution, altitude in triangle, 9-12-15 triangle, right triangle altitude formula."]
