A triangle has sides of lengths 7 cm, 24 cm, and 25 cm. Verify if it is a right triangle, and if so, find its area.
["# Is Triangle with Sides 7 cm, 24 cm, and 25 cm a Right Triangle? Calculate Its Area", "When学习 geometry, one common task is determining whether a triangle is a right triangle and, if so, finding its area. A classic example involves a triangle with side lengths 7 cm, 24 cm, and 25 cm. Let’s explore this triangle to verify its type and calculate its area.", "## Verifying if It Is a Right Triangle Using the Pythagorean Theorem", "A right triangle satisfies the Pythagorean theorem:\nIf (a^2 + b^2 = c^2), where (c) is the hypotenuse (the longest side), then the triangle is right-angled.", "Given sides:\n- (a = 7) cm\n- (b = 24) cm\n- (c = 25) cm (longest side, candidate for hypotenuse)", "Calculate:\n(7^2 + 24^2 = 49 + 576 = 625)\n(25^2 = 625)", "Since (49 + 576 = 625), we confirm:\n[\n7^2 + 24^2 = 25^2\n]\nThis verifies the triangle is a right triangle, with 25 cm as the hypotenuse.", "## Calculating the Area of the Right Triangle", "The area (A) of a right triangle is given by:\n[\nA = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]\nHere, the legs 7 cm and 24 cm serve as base and height.", "[\nA = \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 satisfies the Pythagorean theorem, confirming it is a right triangle. Its area is 84 square centimeters.", "This example illustrates how simple number relationships help identify right triangles and compute their properties efficiently—essential skills for students and enthusiasts of mathematics."]
