A triangle has sides of lengths 7, 24, and 25. Verify if it is a right triangle and find its area.
["# Is a Triangle with Sides 7, 24, and 25 a Right Triangle? Calculating Its Area", "If you’ve come across a triangle with side lengths 7, 24, and 25, you’ve likely noticed something special—this triangle fits a classic mathematical pattern. Whether you're studying geometry basics, solving a problem, or simply exploring triangle properties, understanding whether this triangle is right-angled and finding its area is a great exercise.", "## Is Triangle with Sides 7, 24, and 25 a Right Triangle?", "To determine if a triangle is a right triangle, we apply the Pythagorean theorem, which states that for any right triangle with legs (a) and (b), and hypotenuse (c) (the longest side):", "[\na^2 + b^2 = c^2\n]", "For the given triangle:", "- The sides are 7, 24, and 25.\n- The longest side is 25, so we check if:", "[\n7^2 + 24^2 = 25^2\n]", "Calculating both sides:", "- (7^2 = 49)\n- (24^2 = 576)\n- (25^2 = 625)", "Now add the squares of the two shorter sides:", "[\n49 + 576 = 625\n]", "Since", "[\n625 = 625\n]", "the equation holds true. Therefore, this triangle is a right triangle, with 7 and 24 as the legs and 25 as the hypotenuse.", "## Finding the Area of the Triangle", "For a right triangle, the area is calculated using the formula:", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "Here, the legs 7 and 24 serve as the base and height:", "[\n\ ext{Area} = \frac{1}{2} \ imes 7 \ imes 24\n]", "Multiplying:", "[\n7 \ imes 24 = 168\n]", "Then divide by 2:", "[\n\ ext{Area} = \frac{168}{2} = 84\n]", "So, the area of the triangle is 84 square units.", "## Summary", "- Type: Right triangle\n- Side lengths: 7, 24, 25\n- Right angle confirmed by Pythagorean theorem\n- Area: 84 square units", "This triangle exemplifies a Pythagorean triple—a set of integers satisfying the Pythagorean equation—making it a fundamental example in geometry education and problem-solving. Recognizing right triangles in this way helps students confidently solve triangles, calculate areas quickly, and apply ancient geometric principles to real-world problems."]
