A triangle has side lengths of 7 cm, 24 cm, and 25 cm. Is it a right triangle?
["Is a Triangle with Side Lengths 7 cm, 24 cm, and 25 cm a Right Triangle?", "When studying triangles in geometry, one of the most fundamental questions is whether a given set of side lengths forms a right triangle. A right triangle is defined as a triangle with one angle measuring exactly 90 degrees, and this unique property can be verified using the Pythagorean Theorem.", "Let’s examine the triangle with side lengths of 7 cm, 24 cm, and 25 cm.", "### Understanding the Pythagorean Theorem", "The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side (the hypotenuse). Mathematically, for sides (a), (b), and hypotenuse (c) (where (c) is the longest side):", "[\na^2 + b^2 = c^2\n]", "### Applying the Theorem to the Triangle", "Here, the side lengths are 7 cm, 24 cm, and 25 cm. Since 25 cm is the largest, we assume it is the hypotenuse:", "[\na = 7,\quad b = 24,\quad c = 25\n]", "Now, calculate each side’s square:", "- (a^2 = 7^2 = 49)\n- (b^2 = 24^2 = 576)\n- (c^2 = 25^2 = 625)", "Now apply the Pythagorean condition:", "[\na^2 + b^2 = 49 + 576 = 625\n]", "And compare it to (c^2):", "[\n625 = 625\n]", "### Conclusion", "Since the sum of the squares of the two shorter sides equals the square of the longest side, the triangle with side lengths 7 cm, 24 cm, and 25 cm satisfies the Pythagorean Theorem. Therefore, it is indeed a right triangle — with the 25 cm side being the hypotenuse, and a right angle opposite this side.", "---", "### Why This Matters", "Identifying right triangles is crucial in geometry, architecture, engineering, and physics. Knowing whether a triangle is right-angled helps determine properties like angles, area calculation, and spatial reasoning. For this triangle, you can confidently apply formulas specific to right triangles — saving time and reducing errors in measurements and design.", "TL;DR: Yes, a triangle with side lengths 7 cm, 24 cm, and 25 cm is a right triangle because it satisfies the Pythagorean Theorem ((7^2 + 24^2 = 25^2)). This triangle has one 90-degree angle, confirming its classification as right-angled."]
