Question:** A triangle with side lengths \(7\) cm, \(24\) cm, and \(25\) cm is examined. What is the length of the shortest altitude?
["Title: Find the Shortest Altitude in a Triangle with Sides 7 cm, 24 cm, and 25 cm", "Meta Description: Discover how to calculate the shortest altitude in a triangle with side lengths 7 cm, 24 cm, and 25 cm. Step-by-step solution explained.", "---", "### Understanding Triangles with Side Lengths 7, 24, and 25", "When examining a triangle with side lengths (7) cm, (24) cm, and (25) cm, one key observation stands out: these measurements satisfy the Pythagorean theorem. This tells us the triangle is a right triangle, with the longest side (25 cm) serving as the hypotenuse.", "In a right triangle, the two legs perpendicular to each other form right angles, and the area can be easily calculated. Beyond area, understanding each altitude—especially the shortest—helps analyze geometric properties and solve engineering or design problems.", "### Step 1: Calculate the Area Using Right Triangle Formula", "Since the triangle has legs of (7) cm and (24) cm, the area (A) is:", "[\nA = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height} = \frac{1}{2} \ imes 7 \ imes 24 = 84 , \ ext{cm}^2\n]", "### Step 2: Use Area Formula Involving Altitudes", "The area of any triangle can also be expressed using any side as the base and the corresponding altitude:", "[\nA = \frac{1}{2} \ imes \ ext{side} \ imes \ ext{altitude}\n]", "Rewriting this, the altitude (h) corresponding to a side of length (s) is:", "[\nh = \frac{2A}{s}\n]", "Let’s compute the altitudes for all three sides:", "- Altitude to side 7 cm:\n[\nh_7 = \frac{2 \ imes 84}{7} = \frac{168}{7} = 24 , \ ext{cm}\n]", "- Altitude to side 24 cm:\n[\nh_{24} = \frac{2 \ imes 84}{24} = \frac{168}{24} = 7 , \ ext{cm}\n]", "- Altitude to side 25 cm (hypotenuse):\n[\nh_{25} = \frac{2 \ imes 84}{25} = \frac{168}{25} = 6.72 , \ ext{cm}\n]", "### Step 3: Identify the Shortest Altitude", "Comparing the three altitudes:\n- (h_7 = 24) cm\n- (h_{24} = 7) cm\n- (h_{25} = 6.72) cm", "The shortest altitude is (6.72) cm, corresponding to the hypotenuse (25 cm).", "### Why This Matters", "Knowing the shortest altitude helps in applications where efficiency or space is crucial—such as in structural design, architectural planning, or mechanical components—because it represents the minimal perpendicular distance from the hypotenuse to the opposite vertex. This ensures safety margins and optimal material use.", "### Final Answer", "The length of the shortest altitude in the triangle with sides (7) cm, (24) cm, and (25) cm is (\frac{168}{25}) cm or (\boxed{6.72}) cm.", "---", "Keywords: triangle altitude calculation, shortest altitude formula, area-based altitudes, right triangle altitudes 7 24 25, geometry problem solution, shortest altitude geometry", "Search Intent: Users searching for “What is the shortest altitude in a 7-24-25 triangle?” seek clear, step-by-step guidance to compute altitudes in right triangles and understand their real-world significance.", "---", "Summary:\n- The triangle with sides 7, 24, 25 is a right triangle.\n- The shortest altitude corresponds to the longest side (hypotenuse).\n- Using area ( \left( 84 , \ ext{cm}^2 \right) ), altitude to hypotenuse is (6.72) cm—the shortest altitude.\n- This calculation is vital in engineering, construction, and physics problems."]
