Question**: A triangle has sides of lengths 7 cm, 24 cm, and 25 cm. Is this a right triangle? Justify using the Pythagorean theorem.
["Is a Triangle With Sides 7 cm, 24 cm, and 25 cm a Right Triangle? Prove It Using the Pythagorean Theorem", "Determining whether a triangle is a right triangle can be quickly answered using one of the most famous principles in geometry—the Pythagorean Theorem. This theorem applies specifically to right-angled triangles and is both a powerful diagnostic tool and a foundational concept in trigonometry and Euclidean geometry.", "### What Is a Right Triangle?", "A right triangle is defined as a triangle that contains one angle measuring exactly 90 degrees. The side opposite the right angle is called the hypotenuse, and it is always the longest side of the triangle.", "### The Pythagorean Theorem Explained", "The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides:", "[\nc^2 = a^2 + b^2\n]", "where:\n- (c) is the hypotenuse\n- (a) and (b) are the legs", "If this equation holds true for the side lengths, the triangle is a right triangle.", "### Applying the Theorem to a Triangle With Sides 7 cm, 24 cm, and 25 cm", "Let’s label the sides:\n- Hypotenuse candidate: (25) cm (assumed to be the longest side)\n- Legs: (7) cm and (24) cm", "Now, apply the Pythagorean equation:", "[\n25^2 = 7^2 + 24^2\n]", "Calculate each term:\n- (25^2 = 625)\n- (7^2 = 49)\n- (24^2 = 576)", "Now sum the squares of the legs:", "[\n49 + 576 = 625\n]", "Compare:", "[\n625 = 625\n]", "### Conclusion: The Triangle Is a Right Triangle", "Since the equation (25^2 = 7^2 + 24^2) is true, this triangle satisfies the Pythagorean condition. Therefore, the triangle with side lengths 7 cm, 24 cm, and 25 cm is indeed a right triangle. The side of 25 cm is the hypotenuse, and the other two sides are the legs.", "---", "Real-World Applications\nUnderstanding right triangles and verifying their properties using the Pythagorean theorem is crucial in fields like architecture, engineering, navigation, and computer graphics, where precise measurements and angle identification are essential.", "Use this straightforward method whenever you need to confirm whether a given triangle is right-angled—just square the sides and verify the equality!"]
