Question:** In a triangle with side lengths 7, 24, and 25 units, find the length of the shortest altitude.
["Title: Find the Shortest Altitude in a Triangle with Sides 7, 24, and 25 – Step-by-Step Guide", "---", "Introduction", "Understanding altitudes in triangles is essential in geometry, especially when solving problems involving area, triangle properties, and real-world applications. In a triangle with side lengths 7, 24, and 25 units, identifying the shortest altitude can reveal key insights about the triangle’s structure and area distribution. In this article, we’ll explore how to compute the shortest altitude step-by-step using triangle fundamentals—making it easy to grasp even if you’re new to geometry.", "---", "Step 1: Verify the Triangle is Right-Angled", "First, check whether the triangle with sides 7, 24, and 25 is a right triangle. According to the Pythagorean theorem:", "[\n7^2 + 24^2 = 49 + 576 = 625 = 25^2\n]", "Since the equation holds, this is a right triangle with hypotenuse 25 (the longest side), and legs 7 and 24.", "---", "Step 2: Calculate the Area of the Triangle", "For a right triangle, the area is simply:", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{leg}_1 \ imes \ ext{leg}_2 = \frac{1}{2} \ imes 7 \ imes 24 = 84 \ ext{ square units}\n]", "This area will help us determine each altitude.", "---", "Step 3: Formula for Altitude", "The altitude (height) of a triangle from a vertex to the opposite side is calculated using:", "[\n\ ext{Altitude} = \frac{2 \ imes \ ext{Area}}{\ ext{Base}}\n]", "Since the area is fixed at 84, the altitude depends directly on the base length. The shortest altitude corresponds to the longest side, because the altitude formula is inversely proportional to the base.", "Here, the longest side is 25, so the altitude to the hypotenuse will be the shortest.", "---", "Step 4: Compute the Shortest Altitude", "Use the area and the longest side (25) as the base:", "[\n\ ext{Shortest altitude} = \frac{2 \ imes 84}{25} = \frac{168}{25} = 6.72 \ ext{ units}\n]", "---", "Conclusion: Why This Altitude Matters", "In triangle geometry, the shortest altitude reveals how efficiently the triangle “spreads” its area across its largest side. In this right triangle, the altitude of 6.72 units from the hypotenuse underscores the balance between side lengths and area distribution.", "Always test special triangle properties—like right angles—before applying general formulas. This saves time and increases accuracy in complex problems.", "---", "Summary", "- Triangle sides: 7, 24, 25 (right triangle)\n- Area: 84 square units\n- Shortest altitude corresponds to the longest side (25)\n- Formula: ( h = \frac{2A}{\ ext{base}} = \frac{2 \ imes 84}{25} = 6.72 )", "---", "Keywords: triangle altitude shortest, right triangle altitude 7 24 25, area and altitude triangle, find shortest altitude, 7-24-25 triangle, geometry problem solution", "---", "Vocabulary & Terms:\nright triangle, altitude formula, area of triangle, triangle sides 7 24 25, shortest height geometry", "---", "Meta Description:\nDiscover how to find the shortest altitude in a triangle with sides 7, 24, and 25. Step-by-step guide using right triangle properties and area-based altitude calculations.", "---", "Optimized for search engines with clear headings, keyword-rich content, and practical explanation—this article serves both students mastering geometry basics and anyone seeking to strengthen spatial reasoning."]
