Since \( 7^2 + 24^2 = 25^2 \), it is a right triangle. The area is \( \frac{1}{2} \times 7 \times 24 = 84 \) square units.
["Because ( 7^2 + 24^2 = 25^2 ), It Forms a Right Triangle — Here’s How to Calculate Its Area", "The famous mathematical identity ( 7^2 + 24^2 = 25^2 ) is not just a numerical curiosity — it’s the key to proving that a triangle with side lengths 7, 24, and 25 is a right triangle. Dubbed after the Pythagorean triple (7, 24, 25), this triangle follows the cornerstone of right triangle geometry: the square of the hypotenuse equals the sum of the squares of the other two sides.", "### Confirming It’s a Right Triangle", "To verify the triangle:", "- Square the legs:\n ( 7^2 = 49 ),\n ( 24^2 = 576 ).\n- Add them:\n ( 49 + 576 = 625 ).\n- Take the square root:\n ( \sqrt{625} = 25 ), which matches the third side.", "This proves the triangle has a right angle between the sides of length 7 and 24, confirming it is a right triangle.", "### Calculating the Area: Simple & Powerful", "One of the most intuitive ways to determine the area of a right triangle is to use the two legs as the base and height. Since the triangle has perpendicular sides of 7 and 24 units, the formula applies directly:", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height} = \frac{1}{2} \ imes 7 \ imes 24\n]", "Multiplying:", "[\n\frac{1}{2} \ imes 168 = 84\n]", "So, the area of this right triangle is 84 square units.", "### Why This Matters", "Recognizing a Pythagorean triple like 7-24-25 helps instantly identify right triangles — a foundational concept in geometry, trigonometry, and countless real-world applications such as architecture, engineering, and physics. Coupled with the ability to compute area efficiently, this knowledge makes working with right triangles both intuitive and practical.", "Whether you’re solving textbook problems or designing structures, remembering the 7-24-25 right triangle keeps your geometry skills sharp and your calculations accurate.", "---", "Key takeaways:\n- ( 7^2 + 24^2 = 25^2 ) confirms a right triangle.\n- The area formula simplifies to ( \frac{1}{2} \ imes 7 \ imes 24 = 84 ) sq units.\n- Pythagorean triples are essential for geometry mastery.", "Embrace these tools to unlock clearer, faster math — starting right here in the triangle with legs 7 and 24."]
