A triangle has sides of length 7 cm, 24 cm, and 25 cm. Determine the area of the triangle.
["# Understanding the Triangle with Sides 7 cm, 24 cm, and 25 cm: Calculating Its Area", "Triangles are fundamental shapes in geometry, and recognizing their properties—especially the relationship between their sides—allows us to calculate key measurements accurately. One such triangle with side lengths of 7 cm, 24 cm, and 25 cm stands out due to its perfect adherence to the Pythagorean Theorem, making it a right triangle. This article explores how these dimensions relate, confirms whether the triangle is right-angled, and determines its area using precise mathematical methods.", "## Is It a Right Triangle? Checking the Pythagorean Theorem", "A triangle is classified as a right triangle if the square of the longest side (hypotenuse) equals the sum of the squares of the other two sides—a principle derived from the Pythagorean Theorem:\n[ a^2 + b^2 = c^2 ]\nwhere ( c ) is the hypotenuse.", "For the triangle with sides 7 cm, 24 cm, and 25 cm:\n- Hypotenuse ( c = 25 ) cm\n- Legs ( a = 7 ) cm, ( b = 24 ) cm", "Calculating each term:\n- ( a^2 = 7^2 = 49 )\n- ( b^2 = 24^2 = 576 )\n- ( c^2 = 25^2 = 625 )", "Now test the equation:\n[ 49 + 576 = 625 ] ✅\nSince ( a^2 + b^2 = c^2 ), this confirms the triangle is a right triangle.", "## Why Does This Matter for Area Calculation?", "Identifying a right triangle is crucial because the area can be computed efficiently using the two perpendicular legs—the sides forming the right angle. The standard formula for the area of a triangle is:\n[ \ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height} ]\nIn a right triangle, the legs serve as the base and height. Here, the sides measuring 7 cm and 24 cm are perpendicular, so they are valid for the formula.", "## Calculating the Area Step-by-Step", "Using the identified leg lengths:\n- Base = 7 cm\n- Height = 24 cm", "Apply the area formula:\n[ \ ext{Area} = \frac{1}{2} \ imes 7 , \ ext{cm} \ imes 24 , \ ext{cm} ]\n[ \ ext{Area} = \frac{1}{2} \ imes 168 , \ ext{cm}^2 ]\n[ \ ext{Area} = 84 , \ ext{cm}^2 ]", "## Alternative Confirmation: Area Using Heron’s Formula", "While the right triangle method is straightforward, Heron’s formula offers a comprehensive verification. For triangles that are not right-angled, Heron’s formula estimates area using all three sides:\n[ s = \frac{a+b+c}{2} \quad \ ext{(semi-perimeter)} ]\n[ \ ext{Area} = \sqrt{s(s−a)(s−b)(s−c)} ]", "For sides 7, 24, 25:\n- ( s = \frac{7 + 24 + 25}{2} = \frac{56}{2} = 28 ) cm", "Now compute:\n[ \ ext{Area} = \sqrt{28(28−7)(28−24)(28−25)} ]\n[ = \sqrt{28 \ imes 21 \ imes 4 \ imes 3} ]\n[ = \sqrt{28 \ imes 21 \ imes 12} ]\n[ = \sqrt{7056} ]\n[ = 84 , \ ext{cm}^2 ]", "The result matches our earlier calculation, confirming the area is accurate.", "## Why Knowledge of Triangle Area Is Essential", "Understanding how to find the area of triangles like this one has practical applications in engineering, architecture, graphic design, and everyday problem-solving. Whether designing structures, calculating materials, or solving geometry problems, mastery of triangle properties and area formulas is indispensable.", "## Final Summary", "- The triangle with sides 7 cm, 24 cm, and 25 cm is a right triangle, as it satisfies the Pythagorean Theorem:\n [ 7^2 + 24^2 = 25^2 ]\n- Its area is calculated using the two perpendicular legs:\n [ \ ext{Area} = \frac{1}{2} \ imes 7 \ imes 24 = 84 , \ ext{cm}^2 ]\n- This area can also be verified using Heron’s formula, demonstrating consistency across methods.", "By recognizing right triangles and applying appropriate area formulas, anyone can confidently solve geometric problems involving triangles—turning abstract shapes into practical, measurable dimensions."]
