#### 13Question: A right triangle has legs measuring $ 5 $ cm and $ 12 $ cm. What is the radius of the inscribed circle?
["# How to Find the Radius of the Inscribed Circle in a Right Triangle (Case: Legs 5 cm and 12 cm)", "When dealing with geometry, one of the most important concepts is the inscribed circle (or incircle) of a triangle — the circle that fits perfectly inside the triangle, touching all three sides. For a right triangle, calculating the radius of this incircle is straightforward, especially when you know the lengths of the two legs. In this article, we’ll explore how to find the radius of the inscribed circle for a right triangle with legs measuring 5 cm and 12 cm, and why this method works.", "---", "### Why Find the Radius of an Inscribed Circle?", "The radius of the incircle is vital in various applications, from simple geometric problems to solving real-world engineering and architectural challenges. For a right triangle, knowing this value helps determine the triangle’s most efficient circular fit—important in design, material cutting, and educational assessments.", "---", "### Step 1: Find the Hypotenuse", "First, recall that a right triangle satisfies the Pythagorean theorem:", "[\nc = \sqrt{a^2 + b^2}\n]", "Given legs ( a = 5 , \ ext{cm} ) and ( b = 12 , \ ext{cm} ):", "[\nc = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 , \ ext{cm}\n]", "So, the hypotenuse ( c ) is 13 cm — which is expected from the classic 5-12-13 Pythagorean triple.", "---", "### Step 2: Use the Formula for the Inradius of a Right Triangle", "An important geometric shortcut is the formula for the radius ( r ) of the incircle in a right triangle with legs ( a ), ( b ), and hypotenuse ( c ):", "[\nr = \frac{a + b - c}{2}\n]", "This formula works because the inradius in a right triangle relates directly to the triangle’s area and semiperimeter.", "---", "### Step 3: Plug in the Values", "Using ( a = 5 ), ( b = 12 ), and ( c = 13 ):", "[\nr = \frac{5 + 12 - 13}{2} = \frac{4}{2} = 2 , \ ext{cm}\n]", "---", "### Step 4: Verification Using Area and Semiperimeter", "For completeness, the inradius can also be found via:", "[\nr = \frac{A}{s}\n]", "where ( A ) is the area and ( s ) is the semiperimeter.", "Compute area:", "[\nA = \frac{1}{2} \ imes 5 \ imes 12 = 30 , \ ext{cm}^2\n]", "Compute semiperimeter:", "[\ns = \frac{a + b + c}{2} = \frac{5 + 12 + 13}{2} = \frac{30}{2} = 15 , \ ext{cm}\n]", "Then:", "[\nr = \frac{A}{s} = \frac{30}{15} = 2 , \ ext{cm}\n]", "The results confirm consistency.", "---", "### Summary", "For a right triangle with legs ( 5 ) cm and ( 12 ) cm:", "- Hypotenuse: ( 13 ) cm\n- Inradius: ( \boxed{2} ) cm", "---", "### Final Thoughts", "The radius of the inscribed circle in this triangle is easy to compute using the formula ( r = \frac{a + b - c}{2} )—a handy tool in geometry. Whether you're solving textbook problems or working on real-world design tasks, understanding how to find the incircle radius helps deepen your spatial reasoning skills.", "If you're studying triangles or preparing for math or architecture exams, mastering this concept gives you a strong foundation in geometric relationships.", "---", "Keywords: inscribed circle radius, inradius of a right triangle, 5-12-13 triangle, right triangle geometry, incircle calculation, triangle incircle formula, geometry problem solution.", "---", "Want more geometry insights? Subscribe for regular updates on triangle properties, circles, and mathematical formulas!"]
