A triangle has sides of lengths 7 cm, 24 cm, and 25 cm. Determine its area.
["Understanding the Area of a Triangle with Sides 7 cm, 24 cm, and 25 cm", "Triangles are fundamental shapes in geometry, and calculating their area is a common mathematical task across education, architecture, engineering, and design. One classic example involves a triangle with sides measuring 7 cm, 24 cm, and 25 cm. In this article, we’ll explore how to verify this triangle’s properties and determine its area using precise mathematical methods.", "---", "### Is This a Right Triangle?", "First, let’s check if the triangle with side lengths 7 cm, 24 cm, and 25 cm is a right triangle. According to the Pythagorean theorem, a triangle is right-angled if the square of the longest side equals the sum of the squares of the other two sides.", "Here, the longest side is 25 cm. We compute:", "[\n7^2 + 24^2 = 49 + 576 = 625\n]\n[\n25^2 = 625\n]", "Since both sides are equal, this triangle is indeed a right triangle, with the right angle between the sides of 7 cm and 24 cm.", "---", "### Area of a Right Triangle", "For any right triangle, the area can be calculated easily using the two perpendicular sides:", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "Taking 7 cm and 24 cm as the base and height:", "[\n\ ext{Area} = \frac{1}{2} \ imes 7 \ imes 24 = \frac{1}{2} \ imes 168 = 84 \ ext{ cm}^2\n]", "---", "### Verification Using Heron’s Formula (Optional Confirmation)", "To ensure accuracy and reinforce understanding, Heron’s formula can also be used for triangles with no right angle. However, in this case, since we’ve confirmed it’s a right triangle, Heron’s formula confirms the same result.", "First, compute the semi-perimeter ( s ):", "[\ns = \frac{7 + 24 + 25}{2} = \frac{56}{2} = 28 \ ext{ cm}\n]", "Then apply Heron’s formula:", "[\n\ ext{Area} = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{28(28 - 7)(28 - 24)(28 - 25)}\n]\n[\n= \sqrt{28 \ imes 21 \ imes 4 \ imes 3} = \sqrt{28 \ imes 21 \ imes 12}\n]", "We simplify:", "[\n28 \ imes 12 = 336,\quad 336 \ imes 21 = 7056\n]\n[\n\sqrt{7056} = 84\n]", "Heron’s formula confirms the area is 84 cm², matching our earlier calculation.", "---", "### Why Knowing the Area Matters", "Knowing the area of a triangle like this is essential in real-world applications such as architecture (calculating floor spaces), construction, and design. A triangle with sides 7, 24, and 25 cm reflects real geometric precision and efficiency in structural planning.", "---", "### Conclusion", "The triangle with sides 7 cm, 24 cm, and 25 cm is a right triangle, and its area is 84 cm². Using either basic formulas or Heron’s method provides reliable results, ensuring accuracy in both academic and practical geometry. Whether you’re solving textbook problems or working on real-life design projects, understanding how to compute such areas is invaluable.", "---", "Keywords: triangle area, right triangle area, 7 cm 24 cm 25 cm triangle, Heron’s formula, Pythagorean theorem, geometry guide, how to calculate triangle area."]
