The triangle with sides \(7\) cm, \(24\) cm, and \(25\) cm is a right triangle (\(7^2 + 24^2 = 25^2\)). The area \(A\) of the triangle is:

["The Triangle with Sides 7 cm, 24 cm, and 25 cm: Proving It’s a Right Triangle and Calculating Its Area", "When working with triangles, one of the most fundamental properties is whether the triangle is a right triangle. A right triangle satisfies the Pythagorean theorem: the square of the longest side (hypotenuse) equals the sum of the squares of the other two sides.", "In this article, we’ll explore the triangle with sides 7 cm, 24 cm, and 25 cm, prove it’s a right triangle, and calculate its area using the well-known formula for right triangles.", "---", "### Step 1: Verify It’s a Right Triangle Using the Pythagorean Theorem", "Let the sides be:\n- ( a = 7 ) cm\n- ( b = 24 ) cm\n- ( c = 25 ) cm (the longest side, likely the hypotenuse)", "Check whether:\n[\n7^2 + 24^2 = 25^2\n]", "Calculate each square:\n[\n7^2 = 49,\quad 24^2 = 576,\quad 25^2 = 625\n]", "Add the squares of the shorter sides:\n[\n49 + 576 = 625\n]", "Since ( 625 = 625 ), the equation holds true.\n✅ This confirms that the triangle is a right triangle with the right angle opposite the 25 cm side.", "---", "### Step 2: Calculate the Area of the Right Triangle", "In a right triangle, the area ( A ) is given by:\n[\nA = \frac{1}{2} \ imes \ ext{leg}_1 \ imes \ ext{leg}_2\n]", "Here, the two perpendicular legs are 7 cm and 24 cm. So:\n[\nA = \frac{1}{2} \ imes 7 \ imes 24\n]", "Calculate:\n[\n7 \ imes 24 = 168\n]\n[\nA = \frac{168}{2} = 84 \ ext{ cm}^2\n]", "---", "### Conclusion", "The triangle with sides 7 cm, 24 cm, and 25 cm satisfies the Pythagorean theorem ((7^2 + 24^2 = 25^2)), confirming it is a right triangle. Its right angle lies between the sides of 7 cm and 24 cm.", "The area of this triangle is:\n[\n\boxed{84\ \ ext{cm}^2}\n]", "This elegant combination of geometry and algebra not only proves the triangle type but also makes area calculation straightforward—making it a classic example in mathematics education and practical problem-solving.", "---", "Keywords: right triangle 7 24 25, Pythagorean theorem proof, triangle area formula, right triangle area, 7-24-25 triangle, geometry practice, mathematical verification, area of right triangle."]