A triangle has sides of length 7, 24, and 25 units. Is it a right triangle?
["Is This Triangle a Right Triangle? The Case of Sides 7, 24, and 25", "When studying triangles in geometry, one key question often arises: Is this triangle a right triangle? A powerful way to determine this is by checking whether the side lengths satisfy the Pythagorean Theorem. Today, we examine a triangle with sides 7, 24, and 25 units to answer that very question.", "### What Is a Right Triangle?", "A right triangle is a triangle that contains one right angle (90 degrees). In such triangles, the relationship between the lengths of the sides follows a specific rule: the square of the length of the hypotenuse (the longest side) equals the sum of the squares of the other two sides. This is known as the Pythagorean Theorem:", "[\na^2 + b^2 = c^2\n]", "where (c) is the hypotenuse, and (a) and (b) are the other two sides.", "### Applying the Theorem to Sides 7, 24, and 25", "Let’s assign the longest side—25 units—as the hypotenuse (c), and treat the remaining sides 7 and 24 as legs (a) and (b).", "Calculate:", "- (a^2 + b^2 = 7^2 + 24^2 = 49 + 576 = 625)\n- (c^2 = 25^2 = 625)", "Since (a^2 + b^2 = c^2), the Pythagorean Theorem is satisfied exactly.", "### Conclusion: Yes, It Is a Right Triangle!", "Because (7^2 + 24^2 = 25^2), triangle with sides 7, 24, and 25 units is indeed a right triangle. This means it has one 90-degree angle, making it a classic example often used in geometry lessons and proofs.", "### Fun Fact", "Triangles with integer side lengths satisfying the Pythagorean Theorem like 7-24-25 are known as Pythagorean triples. These naturally occur in stadia, design, and engineering, highlighting the great intersection between math and real-world applications.", "---", "So next time you encounter a triangle with sides 7, 24, and 25, you can confidently say: Yes, it’s a right triangle!"]
