Question:** A triangle has sides of lengths 7 cm, 24 cm, and 25 cm. Verify if it is a right triangle and find its area.
["Verifying if a Triangle with Sides 7 cm, 24 cm, and 25 cm is a Right Triangle and Calculating Its Area", "Triangles are fundamental shapes in geometry, and understanding their properties helps solve a wide range of real-world problems—from architecture and engineering to computer graphics and physics. One common question among students and enthusiasts is: Is a triangle with sides 7 cm, 24 cm, and 25 cm a right triangle? More importantly, can we use this information to find its area? In this article, we’ll explore how to determine whether this triangle is right-angled using the Pythagorean theorem and how to compute its area if it is indeed a right triangle.", "---", "### Step 1: Checking if the Triangle is Right-Angled", "A triangle is classified as a right triangle if the side lengths satisfy the Pythagorean theorem:\n[\na^2 + b^2 = c^2\n]\nwhere (c) is the longest side (hypotenuse), and (a) and (b) are the other two sides.", "For the triangle with sides:\n- (a = 7) cm\n- (b = 24) cm\n- (c = 25) cm", "We identify (25) cm as the hypotenuse because it is the largest measurement.", "Now, substitute the values into the Pythagorean equation:\n[\n7^2 + 24^2 = 25^2\n]", "Calculate each term:\n[\n7^2 = 49\n]\n[\n24^2 = 576\n]\n[\n25^2 = 625\n]", "Now sum the squares of the two shorter sides:\n[\n49 + 576 = 625\n]", "Since (625 = 625), the equation holds true.\n✅ Therefore, the triangle is a right triangle.", "---", "### Step 2: Calculating the Area of the Right Triangle", "For any right triangle, the area can be calculated using the two perpendicular sides forming the right angle. The formula is:\n[\n\ ext{Area} = \frac{1}{2} \ imes a \ imes b\n]\nSubstituting the known values:\n[\n\ ext{Area} = \frac{1}{2} \ imes 7 \ imes 24 = \frac{1}{2} \ imes 168 = 84 \ ext{ cm}^2\n]", "So, the area of the triangle is 84 square centimeters.", "---", "### Why This Matters: Applications of Right Triangles in Real Life", "Right triangles are frequently encountered in construction, navigation, and design due to their predictable properties. For example,? architects and builders rely on the 7-24-25 triangle (a scaled version of the famous 3-4-5 Pythagorean triple) to ensure perfect right angles. Understanding how to verify these triangles and compute their area simplifies a wide range of practical tasks.", "---", "### Summary: Key Takeaways", "- A triangle with sides 7 cm, 24 cm, and 25 cm satisfies the Pythagorean theorem: (7^2 + 24^2 = 25^2), confirming it is a right triangle.\n- The area is calculated using ( \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height} ), giving 84 cm².\n- Recognizing and working with right triangles is essential for both geometry theory and real-world problem solving.", "If you’re ever asked whether a triangle with these dimensions is right-angled, you now have the clear evidence and method to prove it—and even calculate its area with confidence!", "---", "Keywords for SEO:\nright triangle, verify right triangle 7-24-25, area of a right triangle, Pythagorean theorem 7-24-25, triangle area calculation, 7-24-25 triangle, right triangle properties, geometry problems, right triangle example", "Meta Description:\nVerify if a triangle with sides 7 cm, 24 cm, and 25 cm is a right triangle using the Pythagorean theorem and calculate its area—simple steps for students and learners."]
