Check using the Pythagorean theorem: \(7^2 + 24^2 = 49 + 576 = 625 = 25^2\).

["# Check Using the Pythagorean Theorem: (7^2 + 24^2 = 25^2)", "Mathematics is filled with elegant relationships that reveal deep truths about shapes and numbers — one of the most famous being the Pythagorean theorem. This powerful principle tells us that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides:\n[\na^2 + b^2 = c^2\n]", "But why does this work? Let’s explore it using a classic example: checking whether a triangle with sides 7, 24, and 25 is indeed a right triangle — and how the Pythagorean theorem confirms that perfectly.", "---", "## Understanding the Pythagorean Theorem", "The Pythagorean theorem applies solely to right-angled triangles, where one angle measures exactly 90 degrees. The theorem mathematically expresses the relationship between the lengths of the triangle’s sides:", "- (a) and (b) are the lengths of the two shorter sides (legs),\n- (c) is the longest side, the hypotenuse.", "By squaring each side, the theorem shows:\n[\na^2 + b^2 = c^2\n]\nIf this equation holds true, the triangle is right-angled with (c) as the hypotenuse.", "---", "## Applying the Theorem: (7^2 + 24^2 = 25^2)", "Let’s verify this step-by-step:", "1. Compute (7^2):\n[\n7^2 = 49\n]", "2. Compute (24^2):\n[\n24^2 = 576\n]", "3. Add those squares:\n[\n49 + 576 = 625\n]", "4. Compute (25^2):\n[\n25^2 = 625\n]", "Now we see:\n[\n7^2 + 24^2 = 625 = 25^2\n]", "Since both sides are equal, the equation holds true — confirming the triangle satisfies the Pythagorean theorem.", "---", "## Why This Matters: Recognizing Right Triangles", "Using the Pythagorean theorem to check side lengths is not just for homework — it’s a key skill in geometry, engineering, architecture, and navigation. For example:", "- Construction: Ensuring walls and roofs meet at perfect right angles.\n- Surveying: Accurately determining distances across uneven terrain.\n- Game Development & Graphic Design: Calculating distances and movement paths.", "Recognizing valid right triangles saves time and avoids costly mistakes.", "---", "## Bonus: Visualizing the Triangle", "Imagine a triangle with legs of length 7 and 24 units — what does the hypotenuse look like? Since (7^2 + 24^2 = 25^2), you can also visualize a 7-24-25 triangle as a right triangle with clean proportions suitable for construction or tiling projects.", "---", "## Summary", "The Pythagorean theorem is a timeless mathematical truth:\n[\n7^2 + 24^2 = 25^2\n]\nThis confirms the triangle with sides 7, 24, and 25 is a right triangle with hypotenuse 25. Verifying such relationships not only grounds us in geometry fundamentals but also empowers real-world problem-solving across multiple fields.", "Whether you're a student, teacher, engineer, or enthusiast, understanding how to use the Pythagorean theorem is essential — and simple. Test it yourself: anytime you encounter a triangle with sides 7, 24, and 25, you know it’s a perfect right triangle!", "---", "Key Takeaways:\n- Pythagorean theorem: (a^2 + b^2 = c^2) for right triangles\n- Verification of a 7-24-25 triangle confirms it’s right-angled\n- Practical applications in architecture, navigation, and design\n- Visualize the triangle for better spatial understanding", "Keywords: Pythagorean theorem, check right triangle, 7^2 + 24^2 = 25^2, geometry proof, right triangle verification, math proof, geometry fundamentals"]