The triangle with sides 7, 24, and 25 is a right triangle because \(7^2 + 24^2 = 49 + 576 = 625 = 25^2\).
["# The Triangle with Sides 7, 24, and 25 Is a Right Triangle—Here’s How to Prove It", "If you’ve stumbled upon a triangle with sides measuring 7, 24, and 25, you may wonder: Is this a right triangle? The answer is a confident yes—and this claim isn’t just an educated guess. Thanks to the Pythagorean theorem, we can definitively prove it’s a right triangle using basic algebra.", "### What Makes a Triangle a Right Triangle?", "A right triangle is defined by having exactly one angle measuring 90 degrees. The foundational rule for identifying right triangles among any set of three side lengths is the Pythagorean Theorem. This ancient mathematical principle states:", "> In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.", "Mathematically, if (a) and (b) are the lengths of the legs and (c) is the hypotenuse, then:", "[\na^2 + b^2 = c^2\n]", "### Applying the Theorem to Sides 7, 24, and 25", "Let’s test whether 7² + 24² equals 25².", "Calculate each square:", "- (7^2 = 49)\n- (24^2 = 576)\n- (25^2 = 625)", "Now, add the squares of the two shorter sides:", "[\n7^2 + 24^2 = 49 + 576 = 625\n]", "We see that:", "[\n625 = 25^2\n]", "### Conclusion: It Is a Right Triangle", "Since (7^2 + 24^2 = 25^2), the triangle with sides 7, 24, and 25 satisfies the Pythagorean Theorem. Therefore, it must be a right triangle—with the side of length 25 being the hypotenuse, and the angles opposite 7 and 24 being the non-right angles.", "### Why Is This Useful?", "Understanding right triangles isn’t just an academic exercise. These triangles appear in architecture, engineering, navigation, and many STEM fields. Recognizing them quickly helps solve problems involving distances, height calculations, and structural stability.", "### Final Thoughts", "The triangle with sides 7, 24, and 25 serves as a classic example of a right triangle that confirms the power of the Pythagorean Theorem. By verifying the relationship (7^2 + 24^2 = 25^2), we prove that this triangle has a 90-degree angle—making it true that it’s right-angled. Next time you encounter a 7-24-25 triangle, you’ll know exactly why it’s a right triangle—mathematically validated.", "---", "Keywords: right triangle 7 24 25, Pythagorean theorem, right triangle proof, 7^2 + 24^2 = 25^2, mathematical validation, right triangle example, geometry proof, 7-24-25 triangle."]
