Question:** In a right triangle, one leg is 6 cm and the hypotenuse is 10 cm. Find the length of the shortest altitude.
["Finding the Shortest Altitude in a Right Triangle with One Leg 6 cm and Hypotenuse 10 cm", "When working with a right triangle, identifying key geometric elements like legs, hypotenuse, and altitudes is essential for solving problems confidently. One common question in geometry is: In a right triangle where one leg is 6 cm and the hypotenuse is 10 cm, what is the length of the shortest altitude? This article walks through the solution step by step using fundamental triangle geometry.", "---", "### Step 1: Understand the triangle setup", "Given:\n- One leg (let’s call it ( a = 6 ) cm)\n- Hypotenuse (( c = 10 ) cm)\n- The triangle is right-angled, so let the other leg be ( b ), and angle ( C ) the right angle.", "We are to find the shortest altitude of this triangle.", "---", "### Step 2: Find the length of the missing leg", "Using the Pythagorean theorem:", "[\na^2 + b^2 = c^2\n]", "Substitute the known values:", "[\n6^2 + b^2 = 10^2\n]", "[\n36 + b^2 = 100\n]", "[\nb^2 = 64 \Rightarrow b = 8 \ ext{ cm}\n]", "So, the triangle has legs of 6 cm and 8 cm, and hypotenuse 10 cm.", "---", "### Step 3: Calculate the area of the triangle", "The area ( A ) of a right triangle is:", "[\nA = \frac{1}{2} \ imes \ ext{leg}_1 \ imes \ ext{leg}_2 = \frac{1}{2} \ imes 6 \ imes 8 = 24 \ ext{ cm}^2\n]", "---", "### Step 4: Use the area to find altitudes", "The area of any triangle is also given by:", "[\nA = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{altitude}\n]", "We can find all three altitudes—corresponding to each side:", "1. Altitude to the hypotenuse (( h_c ))\nUsing base ( c = 10 ):", "[\n24 = \frac{1}{2} \ imes 10 \ imes h_c \Rightarrow h_c = \frac{48}{10} = 4.8 \ ext{ cm}\n]", "2. Altitude to leg ( a = 6 )\nLet ( h_a ) be the altitude to leg 6:", "[\n24 = \frac{1}{2} \ imes 6 \ imes h_a \Rightarrow h_a = \frac{48}{6} = 8 \ ext{ cm}\n]", "3. Altitude to leg ( b = 8 )\nLet ( h_b ) be the altitude to leg 8:", "[\n24 = \frac{1}{2} \ imes 8 \ imes h_b \Rightarrow h_b = \frac{48}{8} = 6 \ ext{ cm}\n]", "---", "### Step 5: Identify the shortest altitude", "The three altitudes are:\n- To leg 6: 8 cm\n- To leg 8: 6 cm\n- To hypotenuse 10: 4.8 cm", "Clearly, the shortest altitude is the one drawn to the hypotenuse, measuring 4.8 cm.", "---", "### Conclusion: Why the hypotenuse altitude is the shortest", "In any right triangle, the altitude to the hypotenuse is shorter than either leg. This occurs because the hypotenuse is the longest side, and the triangle “spreads out” across it, allowing a relatively short height to span it via area. Thus, the altitude to the hypotenuse of 4.8 cm is definitively the shortest.", "---", "Key Takeaway:\nFor a right triangle with legs 6 cm and 8 cm, and hypotenuse 10 cm, the shortest altitude is ( \boxed{4.8 \ ext{ cm}} ), corresponding to the altitude drawn to the hypotenuse.", "---", "Keywords for SEO optimization:\nright triangle altitude, shortest altitude in right triangle, altitude calculation 6 cm leg 10 cm hypotenuse, triangle geometry solution, find altitude in right triangle, hypotenuse altitude 6 8 cm triangle.", "---", "Understanding how to calculate altitudes empowers you to solve complex geometry problems with ease—start with the Pythagorean theorem, use area formulas, and compare all altitude lengths."]
