A right triangle has legs of length 9 and 12. A circle is inscribed in the triangle. What is the radius of the circle?
["A Right Triangle with Legs of 9 and 12: How to Find the Radius of the Inscribed Circle", "When studying geometry, few problems are as elegant and instructive as finding the radius of a circle inscribed within a right triangle. In this article, we explore a classic case: a right triangle with legs measuring 9 units and 12 units. We’ll walk through the process of determining the radius of the circle that perfectly fits inside the triangle, touching all three sides — the incircle.", "---", "### Step 1: Understand the Triangle", "We’re given a right triangle with legs of lengths 9 and 12. Since it’s a right triangle, we can calculate the hypotenuse using the Pythagorean Theorem:", "[\nc = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15\n]", "So the triangle has sides:", "- ( a = 9 )\n- ( b = 12 )\n- ( c = 15 )", "---", "### Step 2: Formula for the Inradius of a Right Triangle", "For any right triangle, an efficient formula for the radius ( r ) of the inscribed circle is:", "[\nr = \frac{a + b - c}{2}\n]", "This formula comes from combining the area of the triangle and the semiperimeter. Alternatively, it can be derived by noting that the inradius ( r ) satisfies:", "[\nr = \frac{\ ext{Area}}{\ ext{Semiperimeter}}\n]", "Let’s verify and compute using both approaches.", "---", "### Step 3: Compute Area and Semiperimeter", "Area of the triangle:", "[\n\ ext{Area} = \frac{1}{2} \ imes a \ imes b = \frac{1}{2} \ imes 9 \ imes 12 = 54\n]", "Semiperimeter:", "[\ns = \frac{a + b + c}{2} = \frac{9 + 12 + 15}{2} = \frac{36}{2} = 18\n]", "Using the formula:", "[\nr = \frac{\ ext{Area}}{s} = \frac{54}{18} = 3\n]", "---", "### Step 4: Confirm Using the Right Triangle Inradius Formula", "As mentioned earlier:", "[\nr = \frac{a + b - c}{2} = \frac{9 + 12 - 15}{2} = \frac{6}{2} = 3\n]", "Both methods confirm the radius is 3.", "---", "### Why This Matters", "Understanding how to compute the inradius of a right triangle helps in numerous geometric applications, from design and architecture to problem-solving in competitive math. The fact that a simple 9-12-15 triangle yields a clean 3-unit inradius demonstrates the beauty and predictability of right triangles.", "---", "### Summary", "- Given a right triangle with legs 9 and 12, hypotenuse 15\n- Area = 54\n- Semiperimeter = 18\n- Inradius ( r = \frac{54}{18} = 3 )\n- Alternatively, ( r = \frac{9 + 12 - 15}{2} = 3 )", "The radius of the circle inscribed in a right triangle with legs 9 and 12 is 3 units.", "---", "Whether you’re a student learning geometry, a teacher preparing lessons, or a math enthusiast, mastering such problems builds a strong foundation for spatial reasoning and algebraic thinking.", "---", "Keywords for SEO:\nRight triangle inscribed circle radius, inscribed circle formula, inradius of a right triangle, formula for incircle radius, leg lengths 9 and 12, math problem solution, triangle geometry tutorial, right triangle area and inradius, triangle inradius calculation.", "---", "Ready to explore more triangle mysteries? Start with the formulas—your geometry journey begins here!"]
