A right triangle has legs measuring 9 cm and 12 cm. Calculate the length of the hypotenuse and the area of the triangle.
["# Right Triangle with Legs 9 cm and 12 cm – Calculate the Hypotenuse and Area", "When working with right triangles, the Pythagorean Theorem is an essential tool for finding missing side lengths or calculating area. In this article, we’ll explore a right triangle whose legs measure 9 cm and 12 cm, guiding you step-by-step through the calculations of both the hypotenuse and the triangle’s area.", "## Understanding the Right Triangle", "A right triangle has two legs and one hypotenuse, where the hypotenuse is the longest side opposite the right angle. Given the legs:", "- ( a = 9 ) cm\n- ( b = 12 ) cm", "we can use the Pythagorean Theorem to find the hypotenuse ( c ), which states:", "[\nc = \sqrt{a^2 + b^2}\n]", "### Step 1: Calculate the Hypotenuse", "Plug in the values:", "[\nc = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225}\n]", "[\nc = 15 \ ext{ cm}\n]", "So, the hypotenuse measures 15 cm.", "## Step 2: Calculate the Area of the Triangle", "The area ( A ) of a right triangle is calculated using the formula:", "[\nA = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "Here, the legs act as base and height:", "[\nA = \frac{1}{2} \ imes 9 \ ext{ cm} \ imes 12 \ ext{ cm} = \frac{1}{2} \ imes 108 = 54 \ ext{ cm}^2\n]", "Thus, the area is 54 square centimeters.", "## Summary", "- Hypotenuse length: 15 cm\n- Area: 54 cm²", "This triangle example illustrates how basic geometric principles and calculations help solve real-world problems in fields such as construction, design, and navigation. Whether you're measuring a plot of land or planning a DIY project, understanding right triangles is both practical and essential.", "---", "Key Takeaways:", "- Use ( a^2 + b^2 = c^2 ) to find the hypotenuse.\n- Area is calculated using half the product of the two legs.\n- A 9-12-15 triangle is a well-known multiple of the 3-4-5 Pythagorean triple.", "If you’re studying geometry or applying these concepts in technical work, knowing how to compute these values accurately is fundamental."]
