A triangle has sides of lengths 7, 24, and 25. Determine if it is a right triangle and find the area.
["# Is a Triangle with Sides 7, 24, and 25 a Right Triangle? Finding Its Area", "When exploring geometry, one of the most fundamental questions is whether a set of side lengths forms a right triangle. Today, we examine the triangle with side lengths 7, 24, and 25 to determine if it’s a right triangle—and if so, calculate its area.", "## Step 1: Check if the Triangle is Right-Angled", "To determine if a triangle is a right triangle, we apply the Pythagorean Theorem, which states that in a right triangle, the square of the longest side (the hypotenuse) must equal the sum of the squares of the other two sides.", "Given the sides:\n- Side a = 7\n- Side b = 24\n- Side c = 25 (the largest, suspected to be the hypotenuse)", "Check if:\n[ a^2 + b^2 = c^2 ]\n[ 7^2 + 24^2 = 25^2 ]", "Calculate each:\n[ 7^2 = 49 ]\n[ 24^2 = 576 ]\n[ 25^2 = 625 ]", "Add the squares of the shorter sides:\n[ 49 + 576 = 625 ]", "This matches ( 25^2 = 625 ), confirming:\n7² + 24² = 25²", "✅ Conclusion: The triangle with sides 7, 24, and 25 is a right triangle.", "## Step 2: Calculate the Area of the Right Triangle", "For a right triangle, the area is simply:\n[ \ ext{Area} = \frac{1}{2} \ imes \ ext{leg}_1 \ imes \ ext{leg}_2 ]", "Here, the legs are 7 and 24:\n[ \ ext{Area} = \frac{1}{2} \ imes 7 \ imes 24 ]\n[ \ ext{Area} = \frac{1}{2} \ imes 168 ]\n[ \ ext{Area} = 84 ]", "The area of the triangle is 84 square units.", "## Why This Matters", "Understanding right triangles is essential in fields like architecture, engineering, and computer graphics. The Pythagorean Theorem offers a quick and reliable way to verify right angles, while calculating area helps quantify space—critical in design and construction.", "In summary, the triangle with side lengths 7, 24, and 25 is indeed a right triangle, with an area of 84.", "Keywords: right triangle 7 24 25, Pythagorean Theorem, triangle area, right triangle calculation, geometry basics.", "# Is Triangle 7-24-25 Right? Quick Verification & Area Formula", "Whether you're solving math problems or designing structures, confirming right triangles and calculating their areas is second nature with the Pythagorean Theorem. Verifying sides 7, 24, and 25 and finding its area illustrates a core concept that remains vital across science and engineering.", "Start building great geometry skills today—know your triangles!", "---\nRelated topics: Is 7² + 24² = 25²? Area of a right triangle explained. Learn how to verify triangle types and compute area efficiently using the Pythagorean Theorem."]
