E) $ r s = \frac{ab}{2} $ — true
["Understand Why ( r s = \frac{ab}{2} ) Is True: A Clear Explanation", "When you encounter the equation ( r s = \frac{ab}{2} ), it may look unusual at first glance, but it holds meaningful significance in geometry — particularly in the context of triangle areas and the semiperimeter. Let’s unpack this formula and clarify why it is indeed true.", "### The Context: Triangle Area and Heron’s Formula", "The equation ( r s = \frac{ab}{2} ) connects the inradius ( r ) of a triangle, its semiperimeter ( s ), and the lengths of two sides ( a ) and ( b ). Here, ( s ) is defined as the semiperimeter:", "[\ns = \frac{a + b + c}{2}\n]", "where ( c ) is the third side of the triangle.", "One of the most important formulas in triangle geometry relates the area ( A ) of a triangle to its inradius ( r ):", "[\nA = r \cdot s\n]", "This expresses the area as the product of the inradius and the semiperimeter.", "### Linking ( \frac{ab}{2} ) to the Area", "Now consider the area formula for a triangle using two sides and the included angle:", "[\nA = \frac{1}{2} ab \sin C\n]", "where ( C ) is the angle between sides ( a ) and ( b ). This formula naturally arises when two sides and the included angle are known.", "If we assume the triangle formed by sides ( a ), ( b ), and ( c ) has an included angle ( C ), then combining this with ( A = r s ) leads to:", "[\nr s = \frac{1}{2} ab \sin C\n]", "But how does ( r s = \frac{ab}{2} ) arise as a specific identity?", "### A Special Case: Right Triangles and Inradius Formula", "The equation ( r s = \frac{ab}{2} ) becomes particularly meaningful in the special case of a right triangle with legs ( a ) and ( b ), and hypotenuse ( c ). In such triangles:", "1. The area is ( A = \frac{1}{2} ab ), which matches ( \frac{ab}{2} ).", "2. For right triangles, the inradius ( r ) is given by:", "[\nr = \frac{a + b - c}{2}\n]", "3. The semiperimeter ( s = \frac{a + b + c}{2} )", "Now compute ( r s ):", "[\nr s = \left( \frac{a + b - c}{2} \right) \left( \frac{a + b + c}{2} \right)\n]", "This is a difference of squares:", "[\nr s = \frac{(a + b)^2 - c^2}{4}\n]", "Using the Pythagorean theorem ( c^2 = a^2 + b^2 ), expand:", "[\n(a + b)^2 = a^2 + 2ab + b^2\n]", "So,", "[\nr s = \frac{a^2 + 2ab + b^2 - (a^2 + b^2)}{4} = \frac{2ab}{4} = \frac{ab}{2}\n]", "### Conclusion: The Equation Holds in Right Triangles", "Hence, in a right triangle with legs ( a ) and ( b ), the identity", "[\nr s = \frac{ab}{2}\n]", "is true, because it follows from the geometric definitions of area, inradius, and semiperimeter.", "### Practical Takeaway", "While ( r s = \frac{ab}{2} ) is not universally true for all triangles, it holds exactly when triangle ( ABC ) has right angle between sides ( a ) and ( b ). This elegant identity bridges trigonometry, area, and inradius in a simple yet powerful way, illustrating the deep symmetry in Euclidean geometry.", "Use this equation to confirm area relationships in right triangles — it’s a quick geometric reality check.", "---", "Keywords: ( r s = \frac{ab}{2} ), true, right triangle formula, inradius, semiperimeter, area formula, triangle geometry.\nMeta description: Discover why ( r s = \frac{ab}{2} ) holds true in right triangles, linking inradius ( r ), semiperimeter ( s ), and legs ( a ), ( b ) via geometry and trigonometry."]
