Question: A right triangle has legs of lengths $ 5 $ and $ 12 $. What is the length of the shortest altitude?
["Title: Find the Shortest Altitude in a Right Triangle with Legs 5 and 12 – Step-by-Step Solution", "In geometry, understanding the properties of right triangles helps in solving a variety of problems, from theoretical math to real-world applications. One common question is: What is the length of the shortest altitude in a right triangle with legs of lengths 5 and 12?", "This article guides you through calculating the altitudes—specifically identifying the shortest one—using clear step-by-step reasoning.", "---", "### Understanding the Right Triangle", "Given a right triangle with legs of length 5 and 12, the right angle is between these two sides. The hypotenuse can be calculated using the Pythagorean theorem:", "[\nc = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\n]", "So, the triangle has sides 5, 12, and 13 — forming a classic Pythagorean triple.", "---", "### What is an Altitude in a Triangle?", "An altitude is a perpendicular line dropped from a vertex to the line containing the opposite side (or its extension). In a right triangle, two of the altitudes correspond directly to the legs themselves — each leg creates an altitude to the other. However, the third altitude (to the hypotenuse) is shorter and of special interest.", "---", "### Identifying All Altitudes", "1. Altitude to leg 5:\n This altitude is the other leg — 12, since it falls perpendicularly from the opposite vertex.", "2. Altitude to leg 12:\n This altitude is the leg of length 5, equally perpendicular.", "3. Altitude to hypotenuse (13):\n This is the shorter altitude we need to compute.", "Since 5 and 12 are clearly the longer of the two non-hypotenuse sides, their corresponding altitudes are also longer than the third. Therefore, the shortest altitude must be the one drawn to the hypotenuse.", "---", "### Calculating the Altitude to the Hypotenuse", "To find the length of the altitude ( h ) from the right angle vertex to the hypotenuse, we use the area method.", "Step 1: Compute the area using the legs:\nThe area ( A ) of the triangle is:\n[\nA = \frac{1}{2} \ imes \ ext{leg}_1 \ imes \ ext{leg}_2 = \frac{1}{2} \ imes 5 \ imes 12 = 30\n]", "Step 2: Express area using hypotenuse and its altitude:\nUsing the hypotenuse ( c = 13 ) and altitude ( h ):\n[\nA = \frac{1}{2} \ imes 13 \ imes h\n]", "Set the two area expressions equal:\n[\n\frac{1}{2} \ imes 13 \ imes h = 30 \Rightarrow 13h = 60 \Rightarrow h = \frac{60}{13}\n]", "---", "### Comparing All Altitudes", "Now we summarize the three altitudes:", "- Altitude to leg 5: 12\n- Altitude to leg 12: 5\n- Altitude to hypotenuse: ( \frac{60}{13} \approx 4.615 )", "Clearly, the shortest altitude is ( \frac{60}{13} ), corresponding to the hypotenuse.", "---", "### Conclusion", "For a right triangle with legs 5 and 12 (hypotenuse 13), the shortest altitude is the one drawn to the hypotenuse, with length:", "[\n\boxed{\frac{60}{13}}\n]", "This elegant solution combines geometry and algebra to reveal not just the answer, but the reasoning behind it — invaluable for students and math enthusiasts alike.", "---", "### Key SEO Keywords:\nRight triangle altitude, shortest altitude in a right triangle, Pythagorean triangle altitude, triangle altitude formula, altitude to hypotenuse, geometry problem solving, 5-12-13 triangle altitude", "---", "Understanding these principles helps solve not only this problem but future geometric challenges with confidence."]
