Given a triangle with sides \(7 \, \text{cm}\), \(24 \, \text{cm}\), and \(25 \, \text{cm}\), find the length of the shortest altitude.

["Finding the Shortest Altitude of a Triangle with Sides 7 cm, 24 cm, and 25 cm", "In geometry, one of the fundamental tasks is understanding triangle properties, including altitudes. An altitude of a triangle is a perpendicular segment from a vertex to the opposite side (or its extension). When working with a triangle with specific side lengths, identifying the shortest altitude requires careful analysis of area and corresponding base lengths.", "Consider a triangle with side lengths (a = 7 , \ ext{cm}), (b = 24 , \ ext{cm}), and (c = 25 , \ ext{cm}). This triangle is particularly notable because (7^2 + 24^2 = 49 + 576 = 625 = 25^2), confirming it is a right-angled triangle, with the right angle between the sides of 7 cm and 24 cm, and the hypotenuse measuring 25 cm.", "### Step 1: Confirm the triangle is right-angled\nWe verify using the Pythagorean theorem:\n[\n7^2 + 24^2 = 25^2 \quad \Rightarrow \quad 49 + 576 = 625\n]\nSince the equality holds, the triangle is right-angled at the vertex opposite the 25 cm side.", "### Step 2: Compute the area of the triangle\nFor a right-angled triangle, the area is directly computed as:\n[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{leg}<em 7="7">1 \ imes \ ext{leg}2 = \frac{1}{2} \ imes 7 \ imes 24 = 84 , \ ext{cm}^2\n]", "### Step 3: Express altitude in terms of area and base\nThe length of an altitude ((h)) corresponding to a side ((b)) is given by:\n[\n\ ext{Area} = \frac{1}{2} \ imes b \ imes h \quad \Rightarrow \quad h = \frac{2 \ imes \ ext{Area}}{b}\n]\nThis formula allows us to compute altitudes relative to each side.", "- Altitude to side 7 cm:\n[\nh} = \frac{2 \ imes 84}{7} = \frac{168}{7} = 24 , \ ext{cm\n]\n- Altitude to side 24 cm:\n[\nh{24} = \frac{2 \ imes 84}{24} = \frac{168}{24} = 7 , \ ext{cm}\n]\n- Altitude to hypotenuse (25 cm):\n[\nh_{25} = \frac{2 \ imes 84}{25} = \frac{168}{25} = 6.72 , \ ext{cm}\n]", "### Step 4: Identify the shortest altitude\nComparing the three altitudes:\n- (h_{7} = 24 , \ ext{cm})\n- (h_{24} = 7 , \ ext{cm})\n- (h_{25} = 6.72 , \ ext{cm})", "The shortest altitude is (h_{25} = 6.72 , \ ext{cm}), which corresponds to the hypotenuse.", "### Conclusion\nFor triangle sides 7 cm, 24 cm, and 25 cm, the shortest altitude is approximately (6.72 , \ ext{cm}), arising from the longest side (the hypotenuse) being the base. This demonstrates a key geometric principle: the shortest altitude always corresponds to the longest side in any triangle.", "[\n\boxed{6.72 , \ ext{cm}}\n]"]