A right triangle has legs measuring 8 cm and 15 cm. Find the radius of the circle inscribed in the triangle.
["Title: How to Find the Radius of the Inscribed Circle in a Right Triangle (With Legs 8 cm & 15 cm)", "When working with right triangles, one important measurement is the radius of the inscribed circle — also known as the incircle. The incircle is the largest circle that fits perfectly inside the triangle, touching all three sides. For anyone studying geometry or solving math problems, knowing how to calculate this radius is essential.", "In this article, we’ll explore exactly how to find the radius of the inscribed circle in a right triangle with leg lengths of 8 cm and 15 cm.", "---", "### The Triangle in Question", "We are given a right triangle with legs measuring:", "- a = 8 cm\n- b = 15 cm", "Since it’s a right triangle, the legs are perpendicular, and the hypotenuse c can be found using the Pythagorean theorem:", "[\nc = \sqrt{a^2 + b^2} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \ ext{ cm}\n]", "---", "### Formula for the Radius of the Inscribed Circle", "In any triangle, the radius r of the inscribed circle is given by the formula:", "[\nr = \frac{A}{s}\n]", "where:\n- A is the area of the triangle\n- s is the semi-perimeter of the triangle", "For a right triangle, the area is straightforward:", "[\nA = \frac{1}{2} \ imes \ ext{leg}_1 \ imes \ ext{leg}_2 = \frac{1}{2} \ imes 8 \ imes 15 = 60 \ ext{ cm}^2\n]", "The semi-perimeter s is half the sum of all three sides:", "[\ns = \frac{a + b + c}{2} = \frac{8 + 15 + 17}{2} = \frac{40}{2} = 20 \ ext{ cm}\n]", "---", "### Calculating the Inradius", "Now substitute the values into the formula:", "[\nr = \frac{A}{s} = \frac{60}{20} = 3 \ ext{ cm}\n]", "---", "### Final Answer", "The radius of the circle inscribed in the right triangle with legs 8 cm and 15 cm is 3 centimeters.", "---", "### Why This Matters in Real Life", "Understanding the inradius helps in fields like architecture, engineering, and design where space-efficient circles fitting inside boundaries are essential. It also plays a role in optimization problems and physical modeling where contact areas and placement are critical.", "---", "Summary:\n- Right triangle legs: 8 cm, 15 cm\n- Hypotenuse: 17 cm\n- Area: 60 cm²\n- Semi-perimeter: 20 cm\n- Radius of inscribed circle: 3 cm", "If you’re solving geometry problems, remembering this formula — and verifying with area and perimeter — ensures accurate results every time."]
