A triangle has sides of length 7 cm, 24 cm, and 25 cm. Is it a right triangle? If so, what is its area?
["# Is a Triangle with Sides 7 cm, 24 cm, and 25 cm a Right Triangle? Find Its Area", "When it comes to famous triangles in geometry, few are as instantly recognizable as the one with side lengths 7 cm, 24 cm, and 25 cm. This triangle is not only notable for its clean dimensions but also for being a prime example of a right triangle. In this article, we’ll determine whether this triangle satisfies the Pythagorean theorem, and if so, calculate its area in square centimeters.", "## What is a Right Triangle?", "A right triangle is a triangle that has one angle measuring exactly 90 degrees. For a triangle with sides (a), (b), and (c) (where (c) is the longest side), it is a right triangle if and only if:", "[\na^2 + b^2 = c^2\n]", "This fundamental theorem, named after Pythagoras, allows us to verify whether a triangle is right-angled based solely on its side lengths.", "## Checking if the Triangle with Sides 7 cm, 24 cm, and 25 cm is Right", "Given side lengths:", "- (a = 7) cm\n- (b = 24) cm\n- (c = 25) cm (longest side)", "We apply the Pythagorean theorem:", "[\n7^2 + 24^2 = 49 + 576 = 625\n]\n[\n25^2 = 625\n]", "Since (7^2 + 24^2 = 25^2), this triangle is a right triangle.", "## The Special Properties of This Right Triangle", "This particular triangle has integer side lengths—known as a Pythagorean triple. In fact, ((7, 24, 25)) is one of the well-known primitive Pythagorean triples, meaning the sides have no common divisor other than 1. Beyond classification, these triples are essential in solving real-world geometry problems and building practical structures.", "## How to Calculate the Area of a Right Triangle", "One of the great advantages of working with right triangles is that their area can be quickly computed using the two legs (the sides forming the right angle). The formula is:", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "Here, the legs are 7 cm and 24 cm:", "[\n\ ext{Area} = \frac{1}{2} \ imes 7 \ imes 24 = \frac{1}{2} \ imes 168 = 84 \ ext{ cm}^2\n]", "## Conclusion", "The triangle with sides 7 cm, 24 cm, and 25 cm is indeed a right triangle, confirmed by the Pythagorean theorem. Its area measures 84 square centimeters, making it not only a fundamental example in geometry but also a useful shape in fields ranging from architecture to physics.", "Whether you’re studying triangles in school or exploring geometry in real-world applications, this triangle exemplifies clarity, elegance, and functionality."]
