A triangle has sides of lengths 7 cm, 24 cm, and 25 cm. Determine if it is a right triangle, and if so, calculate its area in square centimeters.
["# Is a Triangle with Sides 7 cm, 24 cm, and 25 cm a Right Triangle? Calculate Its Area", "When exploring triangles, one key question often arises: Is this triangle a right triangle? For problems involving lengths of sides like 7 cm, 24 cm, and 25 cm, confirmation as a right triangle unlocks powerful geometric properties — especially when calculating area. In this article, we’ll determine whether the triangle with sides 7 cm, 24 cm, and 25 cm is a right triangle and, if so, compute its area in square centimeters.", "## Understanding Right Triangles and the Pythagorean Theorem", "A right triangle is defined by having one angle exactly 90 degrees, and it satisfies the Pythagorean theorem: for sides (a), (b), and hypotenuse (c) (where (c) is the longest side), the relationship (a^2 + b^2 = c^2) holds true.", "The given side lengths are:\n- (a = 7) cm\n- (b = 24) cm\n- (c = 25) cm (the longest side, likely the hypotenuse)", "Let’s apply the Pythagorean theorem:", "[\na^2 + b^2 = 7^2 + 24^2 = 49 + 576 = 625\n]\n[\nc^2 = 25^2 = 625\n]", "Since (a^2 + b^2 = c^2), the triangle satisfies the Pythagorean relation. Therefore, the triangle is a right triangle with the right angle opposite the side of 25 cm.", "## Calculating the Area of the Right Triangle", "For right triangles, the area can be easily calculated using the two perpendicular sides (legs):", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "Here, take 7 cm and 24 cm as the legs:", "[\n\ ext{Area} = \frac{1}{2} \ imes 7 \ imes 24 = \frac{1}{2} \ imes 168 = 84 \ ext{ cm}^2\n]", "## Summary", "- The triangle with sides 7 cm, 24 cm, and 25 cm is a right triangle because it satisfies (7^2 + 24^2 = 25^2).\n- The right angle is between the sides 7 cm and 24 cm.\n- The area of the triangle is 84 square centimeters.", "Whether you're solving geometry problems or studying for exams, recognizing patterns like the Pythagorean triple (7–24–25) saves time and deepens your understanding of triangle properties.", "Key Takeaway: Always verify the Pythagorean relationship to confirm a right triangle, then use the legs to compute area efficiently."]
