A triangle has sides of lengths 7 cm, 24 cm, and 25 cm. Is this triangle a right triangle? If so, calculate its area.
["# Is This Triangle a Right Triangle? Proving and Calculating Its Area", "If you've come across a triangle with side lengths of 7 cm, 24 cm, and 25 cm, one key question likely arises: Is this triangle a right triangle? Triangles with specific side ratios often fall into special categories, and this particular set is notable due to its exact adherence to the Pythagorean theorem. In this article, we’ll explore whether this triangle is right-angled, how to verify it, and how to compute its area for practical or educational purposes.", "---", "## Step 1: Checking if It’s a Right Triangle", "A right triangle is defined by the Pythagorean theorem: in a right-angled triangle, the square of the hypotenuse (longest side) equals the sum of the squares of the other two sides. Mathematically, for sides ( a ), ( b ), and ( c ) (where ( c ) is the hypotenuse):\n[\na^2 + b^2 = c^2\n]", "For the given triangle with sides 7 cm, 24 cm, and 25 cm:\n- The longest side is 25 cm → this is the hypotenuse if the triangle is right-angled.\n- Let ( a = 7 ), ( b = 24 ), and ( c = 25 )", "Now compute:\n[\n7^2 + 24^2 = 49 + 576 = 625\n]\n[\n25^2 = 625\n]", "Since ( 7^2 + 24^2 = 25^2 ), the triangle satisfies the Pythagorean theorem. Therefore, this is a right triangle with the right angle opposite the 25 cm side.", "---", "## Step 2: Calculating the Area of the Triangle", "For right triangles, area calculation is straightforward:\n[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "In a right triangle, the two shorter sides (legs) act as base and height. Here, the legs are 7 cm and 24 cm. So:\n[\n\ ext{Area} = \frac{1}{2} \ imes 7, \ ext{cm} \ imes 24, \ ext{cm} = \frac{1}{2} \ imes 168 = 84, \ ext{cm}^2\n]", "---", "## Why This Matters: Properties of Right Triangles", "Recognizing right triangles like this one has practical benefits in fields such as architecture, engineering, navigation, and even computer graphics—where precise calculations and efficient geometry are essential. Identifying a triangle as right-angled lets us use its key properties (like area formulas and perpendicular relationships) reliably and efficiently.", "---", "## Final Answer", "✅ This triangle is a right triangle because it satisfies ( 7^2 + 24^2 = 25^2 ).\n✅ Its area is 84 cm², calculated using the product of the two legs over two.", "Whether you're solving a geometry problem, studying for an exam, or designing a structure, knowing whether a triangle is right-angled simplifies calculations and strengthens conceptual understanding. The 7–24–25 triangle is a classic example that reinforces the power of the Pythagorean theorem!"]
