A triangle has sides of lengths 7, 24, and 25. Is it a right triangle? If so, what is the area?
["### Is Triangle with Sides 7, 24, and 25 a Right Triangle? Calculating Its Area", "If you’ve ever explored triangle geometry, you may have come across the well-known triple (7, 24, 25). This set of side lengths raises an important question: Is this triangle a right triangle? And if so, what is its area? The answer is a definitive yes—this triangle is a classic example of a right triangle, and calculating its area is straightforward due to its key right-angle property.", "---", "### Is 7, 24, 25 a Right Triangle?", "To determine whether a triangle with side lengths 7, 24, and 25 is a right triangle, we apply the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) equals the sum of the squares of the other two sides:", "[\nc^2 = a^2 + b^2\n]", "Here, the longest side is 25, so test whether:", "[\n25^2 = 7^2 + 24^2\n]", "Calculate each term:", "- (25^2 = 625)\n- (7^2 = 49)\n- (24^2 = 576)\n- Sum: (49 + 576 = 625)", "Since (625 = 625), the equation holds true. This confirms the triangle is a right triangle, with the right angle between the legs of lengths 7 and 24.", "---", "### Calculating the Area of the Right Triangle", "For any right triangle, the area is simply:", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "Here, the two legs (7 and 24) serve as base and height. Plugging in:", "[\n\ ext{Area} = \frac{1}{2} \ imes 7 \ imes 24 = \frac{1}{2} \ imes 168 = 84\n]", "---", "### Summary", "- Triangle sides: 7, 24, 25\n- Is it a right triangle? Yes, confirmed by the Pythagorean Theorem ((25^2 = 7^2 + 24^2))\n- Area: 84 square units", "This triangle is not only a great visual example but also a useful illustration of right triangle properties in geometry. Its clean, integer side lengths make it popular in educational settings and practical applications alike.", "If you're studying triangles or reviewing geometry basics, recognizing Pythagorean triples like (7, 24, 25) could unlock deeper understanding and confidence in solving related problems. Whether you're calculating area or proving triangle types, knowing these fundamentals is invaluable."]
