A triangle has sides of lengths 7 cm, 24 cm, and 25 cm. Determine if this triangle is a right triangle using the Pythagorean theorem.
["# Is This Triangle a Right Triangle? Exploring the 7 cm – 24 cm – 25 cm Triangle Using the Pythagorean Theorem", "When studying triangles in geometry, one fundamental concept is determining whether a triangle is a right triangle. A right triangle has one angle exactly equal to 90 degrees, and this defining property can be verified using the Pythagorean theorem. In this article, we examine a triangle with side lengths of 7 cm, 24 cm, and 25 cm and determine whether it qualifies as a right triangle.", "## Understanding the Pythagorean Theorem", "The Pythagorean theorem states that in any 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, b, and c are the side lengths with c being the hypotenuse, then:", "[\na^2 + b^2 = c^2\n]", "### Identifying the Largest Side (Hypotenuse)", "For our triangle with sides 7 cm, 24 cm, and 25 cm, the longest side is 25 cm. Therefore, we assume:", "- a = 7 cm\n- b = 24 cm\n- c = 25 cm (hypotenuse)", "### Testing the Pythagorean Condition", "Now, calculate both sides of the equation:", "[\na^2 + b^2 = 7^2 + 24^2 = 49 + 576 = 625\n]", "[\nc^2 = 25^2 = 625\n]", "Since", "[\na^2 + b^2 = c^2 \quad \ ext{(625 = 625)}\n]", "the equation holds true.", "## Conclusion: This Triangle Is a Right Triangle", "Because the sum of the squares of the two shorter sides equals the square of the longest side, the triangle with sides 7 cm, 24 cm, and 25 cm satisfies the Pythagorean theorem. Therefore, this triangle is indeed a right triangle, with the 25 cm side being the hypotenuse and the angle between the 7 cm and 24 cm sides being the right angle.", "Understanding triangle types using the Pythagorean theorem not only enhances geometry skills but also forms the basis for more advanced mathematical concepts. Whether for academic study or practical applications, verifying right triangles is a valuable skill in math and beyond."]
