A right triangle has legs of lengths 9 cm and 12 cm. What is the length of the altitude to the hypotenuse?
["A Right Triangle with Legs 9 cm and 12 cm: Find the Length of the Altitude to the Hypotenuse", "Understanding right triangle geometry is essential for solving many real-world problems in architecture, engineering, and design. One key concept is the altitude to the hypotenuse, which plays a crucial role in calculating areas and solving triangle-related problems efficiently. In this article, we’ll explore a specific right triangle with legs measuring 9 cm and 12 cm and determine the exact length of the altitude drawn to the hypotenuse.", "---", "### Step 1: Find the Hypotenuse Using the Pythagorean Theorem", "Let’s begin with the given right triangle, where the two legs are:", "- ( a = 9 , \ ext{cm} )\n- ( b = 12 , \ ext{cm} )", "The Pythagorean theorem states:", "[\nc = \sqrt{a^2 + b^2}\n]", "Substituting the values:", "[\nc = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 , \ ext{cm}\n]", "So, the hypotenuse ( c ) is 15 cm long.", "---", "### Step 2: Use the Area Formula to Find the Altitude to the Hypotenuse", "An elegant way to compute the altitude (( h )) to the hypotenuse is to use the triangle’s area in two different ways.", "First, calculate the area using the legs:", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{leg}_1 \ imes \ ext{leg}_2 = \frac{1}{2} \ imes 9 \ imes 12 = 54 , \ ext{cm}^2\n]", "Now, express the area using the hypotenuse and the altitude to it:", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{hypotenuse} \ imes \ ext{altitude} = \frac{1}{2} \ imes 15 \ imes h\n]", "Set the two area expressions equal:", "[\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]", "---", "### 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", "Knowing the altitude to the hypotenuse helps in diverse applications like determining the height of triangle-shaped structures, optimizing land use, or calculating forces in engineering contexts. It also strengthens problem-solving skills by combining the Pythagorean theorem and area concepts.", "---", "Try it yourself: Next time you encounter a right triangle, remember: finding the altitude is just a matter of computing area either via legs or hypotenuse + height. Efficient, elegant, and powerful!"]
