A triangle has sides of lengths 7 cm, 24 cm, and 25 cm. Determine if it is a right triangle and find its area.
["Title: Is a Triangle with Sides 7 cm, 24 cm, and 25 cm a Right Triangle? Calculate Its Area Too!", "If you’ve ever explored triangles with specific side lengths, you might have wondered whether a triangle with sides 7 cm, 24 cm, and 25 cm is a right triangle—and how to calculate its area. This triangle isn’t just a shape—it’s a perfect example of a classic Pythagorean triple, making it especially interesting for math enthusiasts, students, and those curious about geometry.", "### Is This Triangle a Right Triangle?", "To determine if a triangle with sides 7 cm, 24 cm, and 25 cm is a right triangle, we apply the Pythagorean Theorem, which states:\nFor a right triangle, the square of the longest side (hypotenuse) equals the sum of the squares of the other two sides.", "Let’s label the sides:\n- Shortest side: 7 cm\n- Middle side: 24 cm\n- Longest side (potential hypotenuse): 25 cm", "We test:\n[\n7^2 + 24^2 = 49 + 576 = 625\n]\n[\n25^2 = 625\n]", "Since (7^2 + 24^2 = 25^2), the triangle satisfies the Pythagorean condition. Therefore, this triangle is indeed a right triangle with 25 cm as the hypotenuse.", "### Finding the Area of the Right Triangle", "One of the most practical aspects of a right triangle is calculating its area effortlessly. Because the triangle’s right angle forms between the two shorter sides (7 cm and 24 cm), these sides form the legs used in the area formula:", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "Here, base = 7 cm and height = 24 cm (or vice versa). So,", "[\n\ ext{Area} = \frac{1}{2} \ imes 7 \ imes 24 = \frac{1}{2} \ imes 168 = 84 \ ext{ cm}^2\n]", "### Key Takeaways", "- The triangle with sides 7 cm, 24 cm, and 25 cm is a right triangle, confirmed by the Pythagorean theorem.\n- Its area measures 84 square centimeters, calculated using the formula for the area of a right triangle.\n- This triangle exemplifies how simple number relationships can reveal powerful geometric truths—perfect for learning, teaching, or simply appreciating the elegance of mathematics.", "If you’re studying triangles, recognizing this Pythagorean triple offers a solid foundation for exploring more complex geometry and trigonometry concepts.", "---", "Try this today: Next time you encounter a triangle with sides near these values, test it with the Pythagorean check—your eye for right triangles will sharpen in no time!"]
