A circle is inscribed in a right triangle with legs of 8 and 15. Find the radius of the circle.
["Understanding the Inscribed Circle in a Right Triangle: Finding the Radius When Legs Are 8 and 15", "When working with right triangles, a fascinating geometric property emerges: the existence of an inscribed circle (also called the incircle) tangent to all three sides. This circle is uniquely determined by the triangle’s dimensions, making it a key concept in geometry and applied mathematics. In this article, we’ll explore how to find the radius of the circle inscribed in a right triangle with legs measuring 8 units and 15 units.", "### The Right Triangle Setup", "A right triangle with legs of 8 and 15 units has:", "- One acute angle (90°)\n- Legs: ( a = 8 ), ( b = 15 )\n- Hypotenuse ( c ), which we calculate using the Pythagorean theorem:\n [\n c = \sqrt{a^2 + b^2} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17\n ]", "So, the triangle has side lengths 8, 15, and 17 — a classic Pythagorean triple.", "### The Formula for the Inradius", "The radius ( r ) of the incircle of a right triangle can be found using a well-known formula derived from the triangle’s area and semiperimeter.", "The area ( A ) of a right triangle is:\n[\nA = \frac{1}{2} \ imes a \ imes b = \frac{1}{2} \ imes 8 \ imes 15 = 60\n]", "The semiperimeter ( s ) is half the perimeter:\n[\ns = \frac{a + b + c}{2} = \frac{8 + 15 + 17}{2} = \frac{40}{2} = 20\n]", "The radius ( r ) of the inscribed circle is given by the formula:\n[\nr = \frac{A}{s}\n]", "Substituting the values:\n[\nr = \frac{60}{20} = 3\n]", "### Why This Works", "In a right triangle, the incircle touches each leg and the hypotenuse, and its center lies at a point equidistant from all sides. Using the area and semiperimeter eliminates the need to directly compute angle measures or use trigonometric identities, making this approach efficient and elegant.", "### Final Answer", "The radius of the circle inscribed in a right triangle with legs 8 and 15 is:", "[\n\boxed{3}\n]", "---", "Key Takeaways:", "- The incircle radius of a right triangle relies on the area and semiperimeter.\n- Using Pythagorean triples like 8–15–17 simplifies calculations.\n- This formula applies universally to right triangles, saving time over coordinate-based or trigonometric methods.\n- Understanding incircle geometry enhances problem-solving in trigonometry, engineering, and design.", "Start finding hidden geometry in everyday shapes — a circle perfectly fitted inside a right triangle awaits discovery!"]
